Preheat grill to medium heat. Cook pasta in pot of water for 6 minutes or according to package.
Cut one dozen 8×12″ pieces of aluminum foil. Layer foil so that you are using 2 per packet. In a bowl, toss together pasta, chopped tomato, basil, salt, pepper, garlic and olive oil. Spoon one serving of pasta mixture (about 1 1/4 cups) into the centre of each packet. Squeeze lemon wedge over pasta before adding the lemon to the packet. Spoon a few teaspoons of goat cheese into packet.
Bring long edges of foil together, then roll it up to secure it. Roll edges of packet to form a closed space (do not squeeze out all of the air).
Reduce grill heat to low, then place packets on grill. Cook for 10-12 minutes, shaking packets halfway through cooking.