Based on a 5 month stay in Brusssels and trips to nearby cities in 2004/2005.
Some notes to prospective European Commission stagiaires
As soon as you know that you are in the "Blue Book", i.e. you passed the preselection, you have to start lobbying aggressively by contacting the heads of your units of interest. You can look up their email addresses and phone numbers in the Commission Staff Directory. Have fun and good luck!
Another useful source of more informal insider information are current stagiaires. Just have a look and see whether any Blue Book stagiaires are listed in your units of interest and contact them. They will generally be more than happy to help you.
Another important issue is where your units of interest are located. Note that part of the Commission is actually based in Luxembourg (which I wouldn't recommend), and some of the Brussels Commission buildings are on the outskirts of the city rather than in the main EU area. Try to find a unit that is located in the EU area, the center of all the action. Current stagiaires or possibly the stagiaire coordinators among the HR staff could tell you roughly where your units of interest would be located (exact addresses of Commission buildings are not available to the general public).
You should also be aware that stagiaires don't have access to their offices on weekends or late in the evening, meaning that you have to bring your own laptop and find a place to stay with Internet access if you are planning on any serious jobhunting.
Housing
My favourite district is Ixelles, particularly the area around Place du Chatelain. The old town is also nice, for example Place Ste Catherine and the Sablon area. For Commission stagiaires, the EU area is a good bet, since many of the stagiaire evening and weekend activities will be happening there (plus you can go running in Parc du Cinquantenaire). Most of the EU area is really ugly, but there are some livable "islands" amid all those office buildings, with old houses, small stores and even a little bit of a "quartier" feel. One is directly north of Rond Point de Schuman but still south of Square Ambiorix (I lived in Avenue Michelange with really beautiful Art Noveau buildings). By contrast, the area north of Square Ambiorix (St Josse) is somewhat dodgy and can be unsafe. Brussels is one of these cities where a few blocks can make a huge difference, so you really should walk around the various neighborhoods to see which parts are fine and which aren't. Another good "island" in the EU area are the streets south of Parc du Cinquantenaire leading towards Place Ste Gertrude.
There are a number of useful links to look for housing such as Praktbxl Yahoo group, The Network Brussels group, Voy group, Expatriates website, Xpats website, Vlanimmo website, Immoweb website. To look up streets, use this Brussels map.
Beware that some unscrupulous landlords specialize on wripping off stagiaires, so have somebody with knowledge of the Belgian legal system at the Commission check the contract before you sign it.
Public transport
Try to be close to a metro station, because buses in Brussels are somewhat irregular with schedules being ignored and trips taking forever due to the crazy traffic in this city. To find out more about itineraries and the like, visit the public transport website. To get around at night or during the frequent bus and metro strikes, a bike can be very useful (just watch out for the generally reckless car drivers in Brussels). You can negotiate a good price for a second hand bike at the Sunday market around Gare du Midi.
Other recommendations
There is a nice litte cultural center near Place Jamblinne de Meux where you can learn African danse, self defense and all that kind of stuff.
Trips from Brussels
The railway system is pretty efficient and there are cheap weekend fares on offer. Since Belgium is so small, go can easily make day trips to other Belgian cities. Ghent, Antwerp and Bruges all have their own distinct character and are well worth a visit. You can also go to Oostende for seafood and then take a tram from there to Den Haan which is probably the nicest seaside town on the Belgian coast. Generally speaking, don't expect too much natural beauty. Most of the coast is pretty industrialized and on weekends, the beach will be crowded like a boulevard. From what I heard, the Dutch part of the coast is supposed to be nicer. If you want to go hiking, head south for the Ardennes.