When in ANY city in Europe, I highly recommend bike tours, ice cream everyday, and afternoon naps. Obviously drinking at local watering holes and eating the regional cuisine are a no brainer as well.
The best thing about a bike tour is that if you do it on the first or second day you arrive, then you've got a tiny glimpse of everything in the city. They are often fast and you normally don't get to stop for pictures everywhere you'd like but then you can go back the next day and spend more time at the things you liked most. In Prague, I took a tour with Praha Bikes. I was SUPER surprised that our tour guide was a 21 year old guy from Austin, Texas. I proceeded to make all kinds of fun of him and undermine his credibility to be giving us a tour of Prague but alas, it was pretty good.
My favorite touristy spots were Old Town Square and the John Lennon Wall. I stayed in Old Town so I walked around there everyday and it's just quintessential Europe. Super busy, lots of street performers, beautiful buildings, and again, ICE CREAM. It has officially become one of my favorite cities in Europe.
The John Lennon Wall is the only place in Prague where graffiti is legal. Because of that, the art changes daily. In addition, there is a street performer playing live Beatles songs in front of the wall. I could've stayed there for hours.
I also recommend spending some time near the bridges and water. Either taking a cruise or just enjoying a coffee. The Charles Bridge is definitely impressive.
However, my favorite part of Prague was definitely the cocktail bars. Originally we set out on a speakeasy bar crawl but it turns out that speakeasies, where the door is hidden or there is a password, is far more popular in the states than anywhere else. Fortunately though, we happened upon three really great cocktail bars in the process. Anonymous Bar and Black Angels Bar are both in Old Town, they had completely different vibes but both unique with delicious drinks. Hemingway Bar is a little more on the outskirts of Old Town but the staff were so much fun and the drinks were extra playful. I could've stayed there for days.
All in all, the sun doesn't set until almost 10pm in the summer so needless to say there is SO much to do and also SO much daylight. Walk around and enjoy!