I'm a huge fan of my A321 Plasma Rifle. It has great damage, great fire rate, infinite range (I'm a master marksman in Fallout 3) and a good clip size.
My favourite weapon was the combat shotgun, that was what i used for most of the game. A pretty boring choice in comparison to some of the other guns but i still loved it. I think the best weapon i used when i played through was definitely the Fat Man.
I'm also a huge fan of the unique Gatling laser, Vengeance. Extremely high damage and fire rate, even bigger clip size, and relatively low AP in VATS. ------- Melee weapon wise. . .probably the Tenderizer. It has good damage, and it is deliciously easy to find more sledgehammers to repair it with.