The level description in the level select screen highlights what the objective is, and there is a objective label over the mission target. Most of the time its going to be in the center of the level. The game uses flight controls so the proper method of turning is to roll (literally roll left or right) and pitch (move the nose up and down) with yaw (rotating left and right) for fine movement. A high G turn lets you turn really fast at the cost of speed, and is integral to completing the game. It can be difficult if your not used to the scheme but it make sense when you do.
Edit: I'll work on getting screenshots up.