I'm just back from a three night stay in Prague, it's a beautiful city and I can't recommend it enough for a city break. Eastern European food though is notorious for being wheat based, with lots of dumplings, wheat thickened stews etc. so I did plenty of research before we went and found some real gems. It's worth taking some basic snacks though if you want to hang out in traditional Czech cafes.
Gluten free is clearly extending out across Europe, I didn't find it particularly hard to eat out whilst there, well, no harder than it is anywhere new.
Firstly, the hotel we stayed in - The Best Western Majestic Plaza offered me GF rolls and cornflakes with the breakfast that was included in our stay. They advised which dishes on the buffet were safe and when the mushrooms weren't one day due to a sauce they were in, they offered and made me a fresh, safe dish of them of my own. They had a really wide selection of stuff, cheeses, sliced sausage, eggs, bacon, fruit, yoghurts. It was excellent. They also tailored meals in their restaurant to suit, and I had a bar snack of baked Camembert with GF bread for dipping.
If, like me, you go to a country and want to try regional dishes, Svejk U Karla restaurant has a full menu of things that you wouldn't normally get, like dumplings, pancakes, apple strudels, the menu was extensive and the prices very reasonable. It's the only 'normal' restaurant I've ever been to where the first menu I was offered without asking was the gluten free one. The potato dumplings were fantastic.
I came across two bakeries that had big, gluten free crossed grain signs in the window and Bez Lepku (Gluten Free) clearly displayed. One is on the Old Town Square and is called Pekarstvi v Dusni. I found part baked bread, biscuits, muffins and cake there. The prices again were a lot less than we pay here (two muffins, some biscuits and two part baked seeded baguettes was about £3.50). The second is Libeřské lahůdky at 9 Vodickova, nearish to Wencleslas Square. I just saw the sign in the window and didn't have time to go in so can't report on what they had for sale.
Our first lunch, at the recommendation of a friend, was at The Bake Shop. It's an upscale deli selling soup, sandwiches, salads and cakes. They were able to do a salad for me with GF bread. I didn't sample any of the biscuits, they were alongside lots of other 'normal' stuff and I didn't want to risk it.
I didn't see any GF products in Starbucks unfortunately. There are plenty of McDonalds around though if you get caught out.
Other places that offer GF menus below, I haven't tried any of these places myself.
Hotel U Prince - this is on the Old Town Square
Little Whale - looks more expensive than many Prague Restaurants
Pizza Coloseum - GF pizza bases and pasta
Triton Restaurant - on Wencleslas Square with an amazing looking cave stalactite ceiling
Another bakery -
Diana Svet Oríšku at Vocelova 606, Praha 2 and other locations - fresh gluten free bread on a Monday
Hope that helps you, and do visit Prague, it's a beautiful city stuffed full of amazing architecture and history.