The equipment needed to begin trad climbing builds on the items from your sport rack. You will still need your harness, belay device, etc. But in addition to the basics of climbing gear, you will need to start acquiring various forms of protection. In modern day, non big wall climbing, there are two main forms of trad protection: passive pro and active pro. A good trad rack contains elements of both.
The first things to buy when you begin building your trad rack are the items necessary to create a top rope anchor. A set of nuts, approximately 10-12 carabiners, and five or 6 slings should be enough if you already have quickdraws on your sport rack. You should also buy approximately 20 feet of accessory cord and some long pieces of webbing or static rope. This should be enough to set an anchor above most climbs, meaning you will be able to access an increasingly larger number of climbs each time you purchase a new item.
You will need to figure out where you will be climbing most often. Different types of rock require different types of protection. Springloaded camming devices(cams) are most effective in an area with high friction and vertical cracks. Hexes and tricams are most often used in areas with a large number of horizontal cracks.
Next you should buy a set of cams in sizes ranging from the equivalent of a Black Diamond C4 #.4 to a Black Diamond C4 #3. Between the cams and the previously purchased nuts, this should be enough to climb many of the shorter single pitch routes in a given area(with the exception of areas known needing more specialized gear). If you are climbing in an area with predominantly vertical cracks you should either double up on sizes .75, 1, and 2 or expand downwards in the sizes equivalent to Black Diamond C3s or Metolious MasterCams, whichever fits your area or style of climbing best.
If you are climbing in an area with an abundance of horizontal cracks or numerous pockets—CAMP tricams become a very useful tool. The are more difficult to place than a cam, but they work in situations that are less than ideal for cams. If a cam stem will be hanging over the lip of a horizontal crack, it is probably better to use a tricam with the sling hanging over the lip instead. They are also the only piece of protection that will work in certain types of pockets. If you get a set of tricams in pink, red, brown, and blue you should be set.
The key to building a trad rack is to make sure each step of the way you are able to access more climbs. By starting with nuts, cord, and webbing you will be able to top rope any climb you can reach the top of. If you add a set of cams to that, you’ll be able to climb most shorter single pitch routes(outside of areas that need more specialized gear, such as tricams.) Once you decide in which direction you will be expanding your rack, you will be able to climb the majority of single pitch routes at any given area. After you have acquired a basic rack, you will quickly learn what supplemental pieces you need for the area in which you climb.