You can't shoot what you can't find. Let's go find something to shoot.
In your lower screen is your radar display. Your position is always in the center of this circular display. The wedge starting from your position and going up is what you can see out of your main viewscreen. The person you are hunting will have a "Hot Box" around them. To bring a target into sight, turn until the "Hot Box" is in the wedge.
The "Hot Box" will only display if the pilot is using "Basic" piloting mode.
If the pilot you are hunting is not on your main viewscreen, a "target direction" arrow will appear on the right or left side of your main viewscreen. Turn in that direction and when that pilot's 'Mech appears in your main viewscreen, the arrow will disappear.
The pilot's 'Mech you are hunting will be outlined in a box on both your main viewscreen (see below) and the radar screen (see above). In an Attrition Mission (Free-For-All), all 'Mechs are fair game, not just the one you are hunting. Therefore you can shoot the 'Mech in the hot box or you can shoot anything else that gets in your way.
When you bring the cross-hairs on top of a BattleMech, The cross-hairs will turn red and the callsign of the pilot will appear below the cross-hairs. Fire your weapons now.