You don't need a trad rack of your own in order to follow. If you're climbing with experienced trad leaders who have their own racks, then you also don't need to bring your own rack. If you're going to lead, you just borrow their gear. In your situation, there is really no advantage to buying a lot of trad gear before you try following on trad climbs at all.
But anyway, to answer your question...
You already have a sport setup, so I'm not going to list stuff like helmet, ATC, etc., that you already have.
You need some passive and/or active pieces. How many and what kind are going to depend a lot on what you climb, and you will get a better idea of this once you've done some climbs and seen what fits in various situations.
You probably want about 10-12 alpine draws for extending your protection. Depending on the climb, it may also be possible to use quickdraws (which you presumably already own) to replace some or all of these, but quickdraws are more of a sport climbing thing, and when you take your course, they'll show you how a rigid quickdraw can pop your protection out of a crack in some situations.
You need materials for building anchors. 1 double-length sling, 1 quad-length sling, 4 locking biners, 10 m (30') of 14 mm (9/16") webbing, some rap rings. Some kind of lightweight knife or scissors.
If you're not already familiar with safety backups for rappels, get some instruction in that, and carry whatever gear you need for the backup, e.g., a prusik.
It's a really good idea to carry a Texas prusik system for use in ascending the rope when you're following on a route that you can't get up.
Nut tool. (And if you don't have one already by the first time you go follow on a trad climb, make sure to ask your leader to lend you one before the climb.)
It's a good idea to at least have a pair of gloves available for when you're belaying a leader. Compared to sport climbing, there is more potential in trad for the leader to take a long fall because something went wrong (pieces pulled out, no good placements for a long stretch,...)
Radios can be handy for situations where communication is difficult. One of the most common causes of accidents is a miscommunication between the climber and the belayer.