I go to Prague several times a year.
Vegetarian is tough, vegan is even tougher. One decent choice is a restaurant called "Country Life" (the name is in English, that's not translated) with three locations around the center.
The day trip recommendations from @Botticelli and @Ed4000 are all great. Another to consider, especially but not exclusively if you like beer, is Plzen: it's the home of the pilsner style, and is a great little town as well.
There are many junk museums in Prague. But a great little one is the "National Memorial to the Heroes of the Heydrich Terror."
An unusual place to check out is "Reon Argondian's Magical Cavern." It's an immersive modern art gallery.
Thanks for the suggestions! I knew that finding restaurants to meet our needs would be tricky, especially after spending some time on yelp, so that I really appreciate the info on the restaurant!
There are some restaurants I've gone to where literally the only vegetarian dish on the entire menu is one fried cheese appetizer. I travelled there some with someone who's pescetarian, which is much easier to accommodate. Typical restaurants have salads and side vegetables, but it's surprising how often those have meat in them, like little bits of bacon.
Staying at an airbnb may be a good option, to be able to prepare some of your own meals.
Yeah, I've been reviewing restaurants and it's going to be tough. I'm okay with picking meat out of stuff, but I just finished chemo, so I am limited on what I can eat until I've been done for a month. This means no salads, uncooked veggies or buffets :( We are staying at an AirBnb, so we will probably eat one meal out and cook the rest back in our room like we usually do. I was looking forward to eating out for each meal since it's so cheap in Prague. Oh well....
Country Life is nice. Some of the by weight restaurants are so inexpensive that I end up going for seconds and filling my stomach.
Slightly out of the way places in Prague: Petrin hill is a nice place to walk off dinner. You can buy a 90 minute transit ticket and likely have time to get to the top, walk down, and hop on another transit to get back to your hotel.
The Museum of Communism is quite amusing. Vysehrad is not in the city's center, but if you have some extra time, it is worth it. We went here - we stayed at Holiday Inn Congress Center, which is not a long walk away.