First of all, some practical stuff:
The Belgian railway network is called SNCB. You could buy separate tickets to go from point A to B, but you can also buy a pass that is valid for multiple trips. The different passes can be found here.
Next, there is an extensive bus and underground network in Brussels. This is called STIB. Here you can also buy passes for multiple trips, these can be found here.
Then, there is a bus network that serves all of Flanders. This is called De Lijn. Info about the passes here. The Coastal Tram also belongs to De Lijn.
There is a seperate bus network for Wallonia too, this is called TEC. Info here (this is only available in French).
It is also possible to buy passes that combine these services, for example SNCB+STIB or SNCB+De Lijn etc.
See how easy Belgium is! :)
There is a lot of stuff to do and see in Brussels alone! Here are just some examples:
- Natural Sciences Museum
- Magritte Museum
- City Hall and Grand Place
- Manneke Pis (it is literally a statue of a little guy that's pissing, our national pride)
I don't know what your preferences are, so I recommend you take a look at this website, where you can also buy a pass that gives you discounts to many museums (Brussels Card). All things in Brussels are very easy to get to, using the bus or underground!
The rest of Belgium:
It is very easy to reach the other cities in Belgium by train from Brussels. Brussels has 3 major train stations: Brussels-Midi, Brussels Central and Brussels North. Brussels Central is right in the city centre and closes to all museums and tourist hotspots. If you are wondering how to get from Brussels to another city by train, simple use the route planner here. This will also tell you the departure times etc. Note that the departure times for trains in the weekends are different than during the week.
- Ostend or Blankenberge (the coast)
These are most of the major cities in Belgium. If you would like to find anything in particular (for example a zoo or you like shopping or you want to hike or rent a canoe and go down a river) just ask!
Just across the border, there are some interesting cities to visit in France. One example I can think of is Lille. But if you have some more spending money, you might even take a high-speed train to Paris from Brussels. There are multiple Thalys trains per day and it only takes 1h30 to get there! Be warned however, these tickets need to be purchased in advance! You can book up to 3 months in advance, how earlier you book, how cheaper the ticket will be.
For more information about international train travel from Brussels, go visit this website
If you want to know anything more specific, you can ask and I'll try and update this answer. There is of course a lot more to Belgium than the things I mentioned here!!