Just for the record, tuna and its relatives are oily fish. But that's apparently not the problem for you I'd thought. Also oily are salmon, all the mackeral, bluefish, trout, dolphin, etc.
Among the common white-fleshed fishes would be halibut, grouper, red snapper, flounder (all the flatfishes, in fact), haddock, pollack, the sea basses, and others, including most fresh-water fishes.
For grilling, first choice goes to the oily fish, for what should be obvious reasons. Among the whites, whole fish work better on the grill than filets; even if they have the skin left on. A whole snapper on the grill, for instance, can be incredible. Filets, on the other hand, are likely to stick and fall apart.
Swordfish also works well on the grill. But, in general, think of swordfish as beef rather than as fish.
You're buddy's suggestion sounds good. I would call it steamed rather than poached (poaching requires submersion in liquid). Cook the veggies to the tender-crisp stage, then add the fish and cover the pan. Just before the fish is done I'd throw in a jigger of bourbon and flame it. But that's certainly not necessary---although salt and pepper is. If this sounds like I make a similar dish it's because I do.
On your tilapia experiment, did the fish sizzle and spit as soon as they hit the oil? If not, the oil was too cold.
Although it's held in disrepute, I still use the drop-of-water-in-the-oil test for pan frying. For deep frying you really need a thermometer. And never fill the pan more than half way with oil.