Within Belgium, take the train. Trains are frequent and the major cities are easily reached. Timetables can be found on www.belgianrail.be, the website of SNCB/NMBS national railway company. From Leuven you have plenty of trains to Brussels, but also hourly connections to places like Ghent, Bruges, Oostende, Liège, Antwerpen or Mechelen.
You can buy tickets per trip. If you do that, note that there are important reductions if you travel during the weekends.
As an alternative, you can also buy rail passes. If you are less than 26 years old, you can buy a Go Pass 10. It costs 50 EUR and you can do 10 trips by train in Belgium. The Go Pass 1 can be interesting too. It costs 6 EUR and is valid for one train trip within Belgium.
If you are more than 26 years old, there is the Rail Pass. It costs 76 EUR in 2nd class and 117 EUR in 1st class. With this ticket you can do 10 trips by train within Belgium.
For some sightseeing trips it is possible to buy a combined leisure ticket, which covers travel and the entry fee.
For a trip to the neighbouring countries, you may also consider the train. You can check schedules and fares at the SNCB Europe website. Also look at the "Offers" tab. You sometimes find there interesting things or hints for trips.
For trips abroad, the bus can be an interesting option too. Eurolines is probably the biggest operator and has lines all over Europe. For travel to Lille and Paris, you may also consider iDBUS. You can also have a look at Megabus if you want to travel to the Netherlands or the United Kingdom.
For travel abroad, you have passes like the famous InterRail pass. Eurolines has a similar system for bus travel. But these are not interesting for weekend trips like those you intend to do. These are for longer trips, and you really have to travel a lot so that they become profitable.
If you choose to travel outside Belgium by train the best strategy is to book as early as you can. This is valid for TGV (France), Thalys (France, the Netherlands, Germany), Eurostar (UK) and ICE (Germany) trains. For the IC trains going to Den Hague, Luxembourg or Lille, you always pay the same, whether you buy a ticket long time in advance or just before leaving. The same is true for the local trains from Liège to Maastricht.
There is nothing like Go Pass or Rail Pass for traveling from Belgium to the neighbouring countries. However there is one trick which can allow you to save some money. This trick applies to the IC and local trains mentioned in the preceding paragraph. You can combine a Go- or Rail Pass with standard tickets. As an example, let's suppose you dispose of a Go or Rail Pass for travel within Belgium. Now you want to travel from Leuven to Luxembourg. You could just buy a standard ticket at the counter or online. However, instead of this you can use your pass up to the last station before the border and then buy an extension to cross the border. With this strategy you can save a few euros, but not strike a big deal. The extension tickets can be bought online or at the station. I recommend the latter option, to be sure that you get what you need ...
If you think that all this is too expensive, have a look hitchhiking or ride-sharing. These are more economical options.