I know where you're coming from, milepig - in North America you don't just hop on a train and go, you have to book your seat in advance.

In Europe, as a general rule, you don't need to book your train travel in advance - you just buy a ticket from A to B and get on the next train that goes from A to B. There are some exceptions (such as the Thalys) but as a general rule seat reservations are optional. Most train companies will, as a matter of policy, only reserve a certain percentage of seats on any given train, which means that there will always be a certain number of seats for those without reservations. You just get on and find an empty seat - if you don't find one you stand until the train starts to slow down for the next station and then you look out for people who look like they're getting ready to get off and then grab their seats the moment they get up.

Chances are that if you buy your ticket the day before travel you'll be able to reserve a seat (late August/early September is the tail end of the summer travel season). If not then the worst that can happen is that you have to stand for part of the journey.

BTW it looks like you can print your own tickets on the SNCF web site - so Paris to Brussels should be no problem (this is a Thalys route so you need to reserve a seat).