dude i dont know if its real tricky to fix it this way, but tuna is supposed to be the best at med rare to rare. ive had it in restaraunts a few times med rare and its pretty damn good. and im a guy that doesnt like a whole lot of red in his meat.
as far as not liking gamey meats. i can understand that. its not for everybody. i for one love lamb chops but i know lots of people that wont eat it. ive even had some fresh goat cooked by some iraqi police that i was patrolling with. pretty tasty.
when my father was working in alaska, he would send me some bison and caribou (reindeer) jerky. i thought the caribou was excellent and it didnt taste too gamey.
Lamb is awesome except for liver. Now that is irony as fuck. Minced lamb>minced beef all day.