[GO TO VYŠEHRAD. I will explain this later, but that's the main piece of advice]
[Do not change money on the street. If they're not con artists, they're finding out where you keep your money so their friends can pick your pockets. I am not kidding.]
Version 1.2. Changes since 1.1: minor changes to the text, added advice for non-beer drinkers (in essence, "see the error of your ways"), added advice on guidebooks, added disclaimer and request for updates. Changes since 1.0: remembered the name of the pub near the embassy, changed to HTML, added Czech letters with diacriticals, added a few new words.
I'd like to keep this guide reasonably accurate, however, so if you've been more recently than me, please send me an email at prague[at]samur41.org.uk (removing spam-defeaters), and let me know what's changed, what you thought was good, and what annoyed you.
Places to stay: the last time I was there, I stayed at a nice place up in Dejvice, in the north-west end of the city, which cost about 200Kč per night. Take the green metro line to Dejvicka, and then head in a roughly north-westerly direction. It was no. 42, I think, but I can't remember the street name. Look on the net, look in guidebooks, you'll probably find it. Out of term-time, you might well be able to stay in student rooms (the place I stayed in was one such): the word for these is "Kolej". I believe you can rent rooms in people's houses for not too much, but I've never done so. The term for that is "penzion", much like in Germany. There's a camp site up in Troja, if that's your thing. Strahov (out in the south-east) is meant to be reasonably cheap. There's probably recent discussion of this issue on the Lonely Planet website message boards, if you can steel yourself to look at them.
A good thing to do when you first arrive is take the tram (no 22 or 26, I think) up to Pražsky Hrad (which simply means Prague Castle), then walk on through the castle, being sure to visit the tomb of Ladislav the Posthumous in the crypt, the window which was used for the defenestrations, and the stained-glass window by Mucha in the cathedral. Then walk down the hill past all the painters into Mala Strana (="little quarter"). See the square with the big pillar with the eye-in-the-pyramid at the top? See the church of Sv. Mikulaš on one side of it? The plain building next to it used to be a seminary, and then used to be an StB listening post, because all the embassies are in this area. Reward yourself for knowing this bit of trivia by having a pint or two in U Hrochu on Thunova street, the British embassy staff's local. (17Kč a pint the last time I was in there, and they don't speak English). Now go into Malostranske Naměsti (Naměsti = "square"), look around, and tick that off the list. Walk out of Malostranske Naměsti in the direction of the Charles Bridge (but call it "Karluv Most" for extra credibility). I'm pretty certain that the John Lennon wall, one of my favourite places, is on the Mala Strana side of the river, on your right as you head towards the bridge. It's definitely near the Maltese Embassy, and a bar that used to be called the Zanzi Bar, (the name may have changed). The wall was a great centre of protest grafitti during the Eighties - someone wrote up "John Lennon RIP" on it, the communists painted it over, someone painted it back up again, and so on, with the protesters soon branching out into more overt anti-communist stuff. It got tagged over during the early Nineties, so when I was living there they painted over the whole thing and started fresh.
When you've seen that, walk across the river over the Karluv Most. Karl, by the way, is Karl IV, who was Holy Roman Emperor, had four wives, and made Bohemia something of a power to be reckoned with - at least, that's the half-remembered Czech version, Clio could give you the real story. He's also the Karl in "Karlovy Vary", the Czech name for Carlsbad. Meanwhile, back at the bridge, note the statues, the extremely worn brass plaque of St John of Nepomuk being thrown into the river (fifteen years ago, it was practically black - the cleaning has all been done by tourists touching it). The statues have all been cleaned recently as well. When you get to the end of the bridge, look to your right. See the disco/club thing there? It's a hole, avoid it. Or at least it was two years ago, though I hear it's just as bad now. The rest of the plan involves making a triangular path encompassing Staroměstske Naměsti (="Old Town Square", with the statue of Jan Hus and the dreaded astrological clock - if you're going to watch it, hang on to your valuables, because Prague is a major pickpocket zone, and the clock striking is feeding time for them), Václavske Naměsti (="Wenceslas Square", though it's actually a very long rectangle), and Naměsti Republiký. There's a nice pub with a beer garden on the right hand side of the road as you walk from Václavske Naměsti to Naměsti Republiký, quite near Naměsti Republiký. When you're in Naměsti Republiký, imagine it full of tanks during the Prague Spring, and when you're in Václavske Naměsti, imagine it full (and I mean full) of people during the Velvet Revolution, gathering to protest peacefully. At the foot of the horse is a small memorial to Jan Palach. The big building at the high end of Václavske Naměsti is the National Museum, which has got some pretty interesting stuff in. If you go about half-way down Václavske Naměsti, take one of the roads leading right, and turn left after a block, you'll get to a nice little tearoom called the Ružova Čajovna (="pink tearoom": "čaj" = tea). I think the street it's in is called Pink Street or something like that. Two more things you should know about Václavske Naměsti: you must have a klobasa from the sausage stands (if you think this is a hardship, you haven't seen the bramboraky), and there's a supermarket and record shop in the metro station at the bottom (called Můstek, which means "little bridge", because there used to be one there).
If you did all that lot, you've seen all the standard tourist stuff, it's only your first day, and you've arrived in Naměsti Republiký in time for happy hour at the Marquis de Sade, which is sort-of in the direction of Staroměstske Naměsti, though it's pretty hard to find. NB there is also a supermarket underneath Naměsti Republiký - go towards Kotva (the big department store), and follow the escalator down.
Other things you should definitely go and see: the Dancing House, which is a twisting glass building by the side of the river (I think it was on the cover of the Lonely Planet a couple of years ago); Vyšehrad, which is the site of the original castle, contains the graves of every important Czech for the last 200 years, and is very very lovely; Naměsti Miru, which is just a nice square with a nice church and the blink-and-you'll-miss-it Čapek memorial; Troja, where I used to live - the zoo's there, but that's a fairly depressing place, there's also a big mansiony thing, a nice walk along the river, and the chance to watch people training for slalom canoe races; the art museum in Holešovice; the funicular railway up to the top of the hill in Mala Strana - climb the tower as well. Naměsti Miru and Vyšehrad have their own metro stops, the art museum is at the tram stop called Nabreži Kapitane Jaroše (or possibly the one after that - it's a huge glass building on your left, you can't miss it. Anyway, try to get a tram that goes over Čechuv Most so you can see the big metronome on the hill). Troja can be reached either by carrying on on the same trams that go to the art museum, or by getting the metro to Nadraži Holešovice and catching bus no. 102 to the zoo, which happens to be next door to the mansiony thing (which is closed on Mondays, I think). If you've got time, have a look at Vystavišťe (though the fountain can probably be missed). Some of the cubist architecture is nice to look at. And go to the House of the Black Madonna for the cubism exhibition and because it's a cool place, (the Czech name is U Černeho Matky Božy, and it's near the Powder Tower), and the Art Deco exhibition at Obecni Dum (the Municipal House) - Prague was one of the centres of Art Deco and Art Nouveau, with Klimt and Mucha and so on. As previously mentioned, there's a stained glass window in the cathedral in the castle by Mucha. Prague has a great variety of architecture, and it's mostly pretty well-preserved, because they weren't bombed in the Second World War. Generally, when walking around, keep looking up: there's lots of decoration, statuary, and so on on the upper floors of buildings, and it's all too easy to miss.
Drinking is dead easy: if you avoid the four big tourist areas (around Malostranske Naměsti, around Staroměstske Naměsti, around Václavske Naměsti and Hradčany) then your beer will set you back 10-20Kč per pint. Food can also be had for 50-100Kč, and usually near the lower end of that, in the pubs (which all have names beginning with "U", which means "At the sign of"). The local beer is Staropramen, which is the best beer in the known Universe: bring me back some and I'll be eternally grateful. Most beer is "světle" or light: dark beer (worth having at least once) is called "tmave". There's a 15th century microbrewery called U Fleku somewhere in Stare Město - great beer, rubbish food. Other wonderful Czech beers are (in descending order of greatness) Krušovice, Budvar, Gambrinus, Pilsner Urquell and Radegast. Several of my more - well, let's be blunt and say "female" - friends have said "but I don't like beer!" before going to Prague. I don't care. Cath went to Prague hating beer and unable to abide even the smell, and came back telling everyone to go to Prague specifically for the beer. I went unable to stand lager, and came back unable to stand only non-Czech lager. Czech beer is a lot easier to get hold of in the UK now (especially since Bass acquired Staropramen and started making a pale imitation of it in Burton), but in its country of origin it's supreme. It is unlike any other beer you have tried. Some evidence for this is the fact that the Czechs drink more beer per capita than any other nation in Europe (though they admit that they'd come second if you counted Bavaria separately). The word for "beer" (pivo) is also the word for "drink".
There are, of course, other drinks available: I'm fond of Becherovka, which is a herbal spirit from Karlovy Vary that's often served as a digestif. "Should you drink Becherovka cold?" we once asked a Czech friend. "Oh yes," he replied, "except for first thing in the morning". A similar (though less nice) drink is Fernet Stock. You will also want to try Slivovice, which is a sort of plum schnapps that usually weighs in at about 60% abv (note that I said "try", not "drink lots of"). And, of course, absinthe. (evil grin). You may think there's little point now that absinthe is legal in the UK, but there are two serious disadvantages to drinking absinthe at home:
Non-alcoholic drinks: there are various tearooms (čajovni) around the place, which tend to serve forty different types of fruit tea but no actual tea. Coffee (kava) is widely available, particularly Turkish coffee. I strongly recommend Viennese coffee (Videňska kava), which is Turkish coffee with a load of whipped cream on top. The cream slowly mixes into the coffee in exciting chaotic patterns, and it tastes lovely. You can also get Irish coffee (Irska kava), and all sorts of other variants on the coffee-with-alcohol-in idea.
If you're going out to eat, there's loads of choice. There's a great place called U Kalicha on Na Bojišti - guidebooks invariably slag it off, but they are very wrong. It's mentioned in the book "The Good Soldier Švejk", and so it's been converted into a Švejk theme pub, which sounds awful, but really isn't: the food's great, and they sing Austrian army songs at you, and the beer's pretty cheap, and they'll love you for not being German. 200-300Kč for the whole lot, and you get a serious amount of food. If you want to be upmarket, the Avalon (near Malostranske Naměsti) and the Nebozizek (top of the hill) are highly posh. If you're in Hradčany and feeling hungry, there's an Indonesian restaurant with meals for about 100Kč. If you're in Dejvice, there's a really nice Italian restaurant on the big roundabout where Dejvicka metro station comes out, and a really nice Asian (mainly Vietnamese) restaurant, where you will feel so underdressed it's not true (but this is not reflected in the price of the meal).
Czech food is basically Austro-Hungarian goulash and meat and stew and so on. I think it's better than the stuff you get in Austria, but I'm biased. Veggie stuff is getting easier to get all the time. The Radost FX and the Terminal internet cafe both had good veggie food. The Czech for "dumpling" is "knedlík", though knedlíky are nothing like British dumplings. Mopping up the sauce from your goulash (sorry, gulaš) with your knedliky is one of the great pleasures in this life. Street food is sausages of various sorts (klobasa and parek v roliku are the names for the big ones and the hot dogs respectively), fried cheese ("smaženy syr") and a really nasty garlicky potato thing called a "bramborak", which will leave you with the taste of grease for at least a week.
Another great advantage of learning Czech is you're suddenly able to pronounce all kinds of things: Unix directory names, number plates... Plurals are usually formed by adding -y to the end, eg klobasy, kluby, ryby. Neuter nouns (ones ending in -o) form plurals with -a, eg piva. But for more than five things, you use the genitive ending, so you should ask for pět piv. Do not worry about this, you'll be understood if you get it wrong, and they'll like you for trying. You're at a slight advantage simply through not being German or Russian.
Hope that helps!