There are three basic ways to rig a plastic worm - Texas, with a lead against the worm head, Carolina with the lead up the line from the worm, and weightless. There are many variations in these three basics, and you have to adjust to the conditions you are fishing, but you should be familiar with all three.
The Texas rig has been around for a long time. A bullet sinker is threaded on the line before tying on the hook. When the worm is threaded on the hook, the sinker sits on its head like a cap - or like the pointed nose of a live worm. This rig slips through brush easily, can be cast under overhangs and through limbs sticking out of the water. It also comes over rocks fairly easily if you use a light lead.
When fishing rocks or very shallow structure, I use a 1/8th ounce lead. That light lead doesn't make a big splash when it hits, comes through the structure without hanging up, and gives the worm a slower fall. If I am casting to brush sticking out of the water or casting under docks and overhanging brush, I stick a toothpick in the back of the lead and clip it off even with the lead. This "pegs" the lead and keeps it from sliding away from the worm as it crawls over brush. It also keeps it from separating if the worm hits a limb or a dock timber and helps it fall straight down.
Deeper structure calls for a 1/4 or 5/16 ounce lead. I hardly ever use a Texas rigged worm with anything heavier. I don't use painted lead, just plain old lead colored ones, but some fishermen swear by the color coordinated leads.
A Carolina rig is when you thread the sinker on the line, tie a swivel below it and then tie a leader for the hook. I usually use a bead or two between the lead and the swivel, especially when using a heavy sinker, to keep the weight from fraying the knot. If fishing rocks, I often put a bead ahead of the lead too, since I think it helps cushion the line and makes it come over the rocks better. The clicking sound the beads make are a plus.
The length of leader can vary from a few inches to several feet. For shallow water I usually stick with about 18 inches. This shallow water rig has a 3/8 or 1/2 ounce sinker and is on fairly light line. I like to cast it around underwater brush and short bottom cover. If the grass or other cover is tall, I use a leader long enough for the worm to be near the top if it.
When fishing deeper, I use a 30 inch leader and a bigger lead, usually 3/4 ounce. This rig is good for "raking" gravel points and banks with little cover. If I want to fish fast, I go with a one ounce sinker. In deeper water you can get by with heavier line but I often use a lighter leader to make sure I don't spook the fish.
A variation of the Carolina rig is a split shot rig. A small split shot is crimped a few inches up the line from the worm. For some reason, maybe its small lead size, this rig will draw strikes when others will not. Throw it around grass, brush and docks. It sinks slowly and usually you will see your line start to move off without feeling a strike.
Weightless worms have really made a comeback recently. We used them back in the 1950's and 60's and they came rigged with two hooks and a spinner in front. They caught fish - and still will. Now, many companies make a worm like the Zoom Trick Worm in bright colors. It is rigged weightless and fished on top or very shallow. You watch the worm, looking for a strike, and that is why you need a bright color. The fish seem to like it, too.
Choice of color, size and style of plastic worms vary tremendously. Maybe that will be a good discussion for later.