I went around two years ago, though not for as long as you're going (it was the end of me and my girlfriends little travels in Poland and Czech Republic involving Warsaw, Krakow, Brno and Prague).
The stuff we did was:
- A tour of the city on a riverboat and we joined a group to get a walking tour of the old town. This was fun, learned about the city and you get a great view of the bridges and buildings along the bank (and the penguins in the river

).
- The castle, though only the parts which were free such as the gardens and
St Vitus Cathedral. So we didn't get to do the Golden Lane.
- We did pay to go into the toy museum in the castle though, as it was cheap and has one of the biggest collections of barbie dolls in the world (if you go make sure to find the Donald Trump barbie doll!) along with loads of old toys from loads of different times and some Action Men and GI Joes (including an astronaut one which me and my brothers had

).
- We did pedalos on the river which was fun. You can get some good pictures of the wonderful buildings along the riverbank. I also got complimented on my choice of footwear brand (Meindl) by one of the pedalo company people
- Charles Bridge. Loads of street artists and musicians, can get very crowded.
Even though Prague is more expensive than other parts of the Czech Republic, food and drinks are still pretty cheap as long as you can avoid the tourist traps. You should definitely try the local czech beers, such as Staropramen and Pilsner Urquell. These are cheap and, in my opinion, very good.