Trim stems from beets to 1 inch; cut into 1/2-inch-thick rounds. Cut bell peppers into 1-inch strips. Cut off ends of zucchini and eggplant; cut each lengthwise into 3/4-inch-thick slices. Cut onion into 1/2-inch-thick rounds.
Whisk together oil and vinegar in a large bowl. Add cut vegetables, and sprinkle vegetables evenly with salt and pepper, tossing to coat.
Grill vegetables, covered with grill lid, over medium-high heat (350° to 400°). Grill beets 4 to 7 minutes on each side or until tender; grill bell peppers, zucchini, eggplant, and onion 3 to 5 minutes on each side or until crisp-tender. Cool beets slightly, and, if desired, peel. Sprinkle vegetables with goat cheese, if desired.