You want a really firm-textured fish for grilling. Tuna, swordfish, shark, halibut, etc., will work. Salmon is great when grilled, but, if mishandled, can fall apart. More flakey, tender fish, like Tilapia, Cod, etc., should be cooked by a different method.
Tuna is best when cooked to medium. It's hot, juicy, and full of flavor. I haven't had Swordfish, so I can't give any advice. Shark is very strongly flavored, and benefits from a brief marination in milk to remove some of the "fishy" taste.
I wouldn't use a pair of tongs to turn the fish, as this makes it easy to damage the fish. A nice, thin, flat metal spatula works wonders.