I want to buy all of the Halo games for the 360 on disc, but do I need free space to install them? I've never had an Xbox before.
You don't, in general, have to install disc games to play them on an Xbox 360, although it will often make the load times faster.
Some games do require installation, but the Halo games are not among those. Indeed, Halo 3 should not be installed, because it's optimized to run faster from the disc.
You do, however, need to have a little free space on your hard drive in order to create save data.