First rule of engagement in my opinion is concentration of fire (unless you play Clan by Zellbringen rules). Try to utilize cover and position your units along avenues of fire so that as many of your own units have decent shots as possible on a single enemy unit, while the other enemy units are blocked from firing at your own units. As long as an enemy isn't shooting at any of your units, it's not contributing to the fight.
On the other hand, positioning a unit for the next turn, so that it can maneuver to get an ideal shot (such as a backshot on a HBK) on the next turn can sometimes pay off handsomely, even if it means not being able to fire on the current turn. Positioning for next turn also means leaving yourself with more than one movement option, so if things suddenly get ugly, you're not committed to edging down a steep hill toward an opponent that has just pulled an SRM Carrier out from behind the woods where it was hidden, and where turning around to go back UP the hill would expose your rear armor.
As pointed out, keeping to ranges that favor your own weapons more than those of the opponent means better to-hit numbers, increasing the odds of doing more damage than you receive. Calculating movement to maximize the to-hit number difference between your own shots and the opponent's again skews the odds in your favor. Be aware of minimum ranges: running a light unit up to 2 hexes from an LRM Carrier means TERRIBLE odds for the opponent to hit you with its main weapons (and minimal or no secondary weapons), while your own short-range weapons (such as Medium Lasers and SRMs) have short-range shots. Backshooting an AWS, as another example, can be quite effective, as the 'Mech has only an arm-mounted PPC that it can turn back toward you, and that has a minimum range that will render that single shot a lot less likely to connect. While you're just standing there loitering for the rest of the turn, a kick or punch can ruin their day if it causes a fall or hits the right location. Of course, if the enemy has a second AWS standing 4-5 hexes from the first, you might be in short range of a weapon you'd rather not have pointed in your direction.
Of course, the most important rule in Battletech combat is: roll well. A good die roll or two can easily turn the tide of an otherwise poorly played game.