When first closing to within sensor range of the enemy... Sprint your mechs to cover (but not so close that you establish visual contact with the enemy). The sprint will give you high evasion against the enemy's first attacks. Combat mode will begin on the next round after sensor contact is made. If you accidentally make visual contact, combat begins immediately, and your units that have already moved do not get to move again. On the first combat round, reserve your units for as long as possible, letting the enemy move into visual range.
Once you're attacking, most of your mechs will want to move far enough to get some evasion bonus and still end up in cover. This reduces the damage that you will take. Warriors with the guardian skill get buffs for not moving. For everyone else, "speed is life", since it gives you evasion >>> arrows.
Prioritize targets based on their squishiness and the amount of damage or stability damage that they cause. LRM and SRM carriers should be destroyed with extreme prejudice because they deal more stability damage than you'll want to deal with. Strikers are easy to squish, and also get priority from me due to their missile attack. Remember that vehicles are more vulnerable to stomping melee attacks than mechs are. After that, pick the nearest mech with the least armor to target (or one that has low armor for its attack capability like the -D version of the Shadowhawk).
Use a precision strike against the closest leg of the target mech. If one of your warriors has the "High Spirits" buff, it costs half as much morale as normal to make this attack.
Concentrate your fire on the chosen target. Have your other units attack the same mech from the same side. Even if you're not using precision strike, attacks from the side will frequently hit the leg. Blowing a leg off of a mech knocks it down and gives it a -1 initiative penalty. Use your weapons that cause the most stability damage. Even if you can't take off the leg in one round, knocking the mech down with stability damage is almost as good. Because the enemy mech regains some stability each time that it moves, this is the reason for concentrating your entire lance's stability damaging fire on one target at a time.
Once the mech is knocked down, you get free called shots from all of your remaining units until that mech moves again. If your target had already moved this round before you knocked it down and you've reserved movement of your lighter mechs until later initiative phases, you will get to attack this round and then again in the early initiative phases of the next round before the enemy gets his turn to stand up. Target the legs with your free called shots. If the one-legged enemy stands up, use a precision strike on the remaining leg.
Once the mech is legless, it is rendered hors de combat. You'll normally have the opportunity to salvage two out of the three possible pieces of the mech chassis. If your called shots were accurate, you've also probably not destroyed as many valuable weapons and equipment components as you would if you destroyed the left and right torso of the mech.
1: i7-9700K, NH-D15, Z390M Pro4, 32GiB, RX Vega64, ½TB 960Pro +2TB MX500, Define Mini-C, SSR-850PX, LG 32UD99 + UltraSharp U2410, RK9000V2-BR
2: R5-3600X, Wraith Spire, B450M Steel Legend, 32GiB, RX5500XT, 2TB SX8200Pro, P160, SS-660XP2, TV, G610