After living in Praha 7 (Holesovice, just north of the bend in the Vltava) while studying abroad in Prague, I can honestly say that it is my favorite city in Europe (though I guess I'm a bit biased). While in East Central Europe, I also had a chance to visit Krakow, Bratislava, and Budapest, as well as other Czech cities and some cities in both NE and SE Germany.
Although Prague is terribly crowded in the summer, I'd still recommend stopping by. Just wandering aimlessly around the streets of Old Town is an amazing experience, especially if you're aware of your surroundings (i.e. for pickpockets) and don't let the crowds get to you. You can see Prague Castle, Charles Bridge, Wenceslas Square, and Old Town Square in about half a day. After that, you can escape the intense tourist crowds and see other parts of the city that are less well known, such as Troja Chateau (by the zoo, north of Stromovka Park and the 2nd Vltava bend), a beautiful summer home which now houses 19th century art, and Veletr×nÝ Palace, the section of the National Gallery which houses the modern art collection. Fittingly, it is in a communist-style building in a quiet neighborhood in Holesovice. I'd also highly recommend to art fans the riverfront St. Agnes' Convent, which has the Medieval collection of the National Gallery. Strahov Monastery with its incredible library the Loreto are well worth visits if you're looking to avoid the massive crowds of Old Town and Castle Hill (there still will be some tourists in those areas, however). If you enjoy nature, I'd recommend Letna Park, which has nice views, especially at night to Old Town across the river, as well as any of the Islands in the Vltava. For everyone, Petrin, a hill overlooking the city with a minature replica of the Eifel Tower, is a can't miss. Make sure to go on a clear day because the views are incredible.
If crowds aren't your thing, I'd encourage you to explore Moravia. Its capital, Brno, is quite a pleasant city and a good point from which you can go on many day trips. Brno actually reminds me a little of Bratislava, but without all the modern art and without the Communist bloc feel. The nearby Moravsky Kras region has tons of underground caves and rivers. It was honestly one of my favorite things I did in the entire region.
Even though Cesky Krumlov (in Bohemia about 3 hours south of Prague) is beautiful, it is also becoming quite crowded and touristy. If you venture further east, however, you'll discover many hidden treasures, where most of the tourists are Czech school groups rather than Americans and Aussies. I'd highly recommend KroměřÝ×, a UNESCO World Heritage Site, which was the seat of an Archbishop. Today you can visit the Archbishop's palace, gardens, and wine cellars (which for centuries made the Communion wine for most of Eastern Europe). Another UNESCO site in Southern Moravia is Lednice, where there is a castle from the House of Liechtenstein (that family used to rule most of the Czech Republic before the Habsburgs). A beautiful English-style garden with a Minaret surrounds the palace.
I've noticed that there have been a lot of comparisons to other East Central European cities as well, so I might as well throw in my two cents. Bratislava was alright. I just took the night bus down one evening, arriving at around 7am, and spent an entire day there, taking the night bus back at midnight. After about 6 hours hanging around old town, I was already bored and retracing my footsteps. I ended up sitting on one of the benches in the lovely tree-lined area by the national theatre and reading for three or four hours. I did enjoy looking at the random modern art statues around the city, though. I was highly looking forward to visiting Budapest because I'd heard wonderful things about the "Paris of the East". I found the city dull and gloomy. The Castle Hill wasn't nearly as pretty as that of Prague and the city was much more polluted, almost giving it a hazy appearance. The Turkish-style baths that we'd looked forward to visiting were disappointing. They were terribly overcrowded and reminded me more of an American amusement park than the other hot steam baths I've visited in Eucador and Japan. I would, however, highly recommend Krakow, easily one of my top five European cities (number two to Prague in this region). It had a young and exciting vibe with an upbeat shopping district and lively nightlife. Market Square had a great feel to it, though it was crowded and touristy. I'd highly recommend visiting Kazimierz, the Jewish district, which is about a 15 minute walk from the ring-street inner city. It is a sobering experience, but I feel like learning at least a little about the Jewish history of Poland is a must if you're there.
Hope this was at least somewhat helpful! Feel free to look at my travel blog or e-mail me if you have any questions or want Prague suggestions!