Hey! Here is my list for you! 5 days is a good amount of time in Prague. It's a small city but there are a lot of different things to do.
Of course, you need to do the basic stuff, like walking in the small streets of the Old Town and going the the Old Town Square where the Astronomical Clock is. Then there is the Charles Bridge no far from there but I advise you to also look at it from another bridge! When you cross the bridge, you are close to the Prague Castle (it's huge but personally I don't really like to go there and find it more beautiful from far away :) ).
And when you walk to go to different places, do not hesitate to go inside some small venues. Even if they don't look nice on the outside, it can be very nice inside. That's also usually the cheaper places.
Now, I can recommend you some less known places. The first one is the Letna Parc. It's in Prague 7 and you have a beautiful view of all Prague from there (there ara also some bars to have a beer).

Then there is National Monument of Vitkov in Prague 8 (I'm not sure if it's Prague 8 but you will find it). You can also go to the Petrin tower. The tower is not really great but it's a really peaceful place without the noise of the city.
And of course, it's currently the time of the Christmas Markets! There are 5 or 6 different markets in the city. The bigger is on Old Town Square. It's beautiful but very crowded during the weekends. I like the one on Wenceslas Square (even if it's in the center, there are not a lot of people). There is also a good one on Namesti Miru. Personally I don't find the one on Namesti Republiky really great.
I'm studying here so I also know a lot of bars! Here some of them that I like:

• U Sudu Bar (close to the center): it seems small but there are lot of different rooms in the basement. Even when it's packed, the service is fast and the beer is very good (it's Pilsner Urquell).
• Scout Institue (Old Town Square): it's a hidden bar 50 metes away from the astronomical clock. You will have to find it ;). I like it because it's in the center and all the tourists walk in front of it without knowing. And moreover the beer is cheap for the place (40 czk I think).
• Cobra (in Prague 7): even if it's not in the center, there are always people in this bar (from 9 am to 2 am). You can also eat there.
• Dog Bar (the real name is Vzorkovna): this bar is really special but it's worth a visit. I can't explain it but you probably never see a bar like that!
• Chapeau Rouge (the bar part and not the club): the mood is very nice there and it's well decorated.

If you like to go to the club I can also recommend some of them but the number one rule is to avoid the 5 story club Karlovy Lazne.

For the restaurants, most of the ones I know are situated in Prague 7 because I live there. I know a good one in the center. It's called U Provaznice and it's cheap for the location. Try the Gulasch or the Svickova and a normal price is approximately 150-200 for a meal like this (so try to avoid tourist trap with high prices).

And to visit the city I recommend you to take a pass for the public transportation, they are everywhere even during the night. And don't take taxis, they will probably rip you off. If you need to go somewhere by car, use Uber or Taxify (similar to Uber but you can pay cash).

I hope these advice will be useful for your trip. If you have question, do not hesitate to ask :)