A lovely city
While everything is fairly close together, and within walking distance as you say learning where things are is a good idea otherwise you'll be doing an awful lot of walking. The Charles Bridge connects the two sides of the city well around the touristy bits, so it's worth learning whereabouts it is! There doesn't seem to be a great deal of public transport around the tourist areas as it's old, thin cobbled streets so we just walked everywhere. Maybe somebody else that's been may feel differently.
On the one side of the river you have the castle etc, definitely worth a visit just be careful which tickets you buy for the castle. You can get into the inside for free, then pay for each section you want to go in. Or you can buy what you think is everything only to find you have to pay extra to go up the tower, like we did
On the same side, a little further round is Petrin tower which is also good, a nice walk from the castle to the tower through the park. If you aren't good with heights though I would be wary of going up the tower, the stairs are outside, and it sways in the wind
brilliant views of the city from up there though!
On the eastern side of the river is the Old Town Square where lots of the walking tours start from, many of them free and you just tip your guide at the end, just look for the umbrella with your language on! There are also lots of places to eat here, although on the square is dearest if you go along the side streets they tend to be a little cheaper. Also on the square is the famous clock, make sure you catch the clock on the hour. Don't worry, there will be a crowd! Up towards the river again from the Old Town Square is the old Jewish quarter, which is now designer shops etc. However there are still some synagogues that are worth a visit as they're memorials and museums to the Jewish population that lived there before the Nazi's arrived, the Spanish synagogue is particularly impressive. Photography isn't really allowed in them though, even with a photography ticket (which you pay extra for) but worth the experience. If you wanted a walk to Letna Park it is up this way also, just walk over the bridge from the Jewish quarter and you're there. There are also a couple of boat tours that leave from up here.
Further over on the eastern side is Wencesles square, where the shops are but there's a large museum at the top, it was closed for refurbishment when we were there so can't comment! if you google the Velvet Revolution, you'll see lots of the images are from here.
There are lots of opportunities to get above the city in the old towers which cost a couple of pounds a time but give brilliant views. If you were only going to do one, make sure it's the one on the eastern side of the Charles bridge
If cocktails are your thing, we found a nice little place called Hemingway bar, small little quiet place with some really good bar staff. The kind of place with a menu but would make you anything you wanted. A good place to stop for a bite during the day is a place called the Grand Orient Cafe, Art Deco style and brilliant food but at cheap prices, the strudel was superb
20170217 - View from the Tower - Hotel Paris Prague by Chris Mitchell, on Flickr
