It’s also worth noting that if you’re struggling with a SRM/LRM heavy lance you are probably not focusing on flank shots on individual 'Mechs enough. LRMs are barely a damage source - instead, they are a decent source of Stability Damage, which makes 'mechs fall over. This is a tactic that has its place (especially for salvage, or temporary threat mitigation).
I would also worry about ammunition on the longer story missions. I quite like SRMs but I don’t depend on them.
'Mechs have 11 different independent armor locations and the problem with SRM/LRM volleys is that the individual missiles will be randomly allocated between the available armor. If you’re hitting from the rear arc, that’s good, because there’s only 3 armor panels. From the side, also not too bad, with 4 armor panels to hit, and they’re all high-value panels. From the front, it’s pretty bad, because missiles fired at the front arc will chip away at eight armor locations, and one of them is an extremely rare one to hit. This means that instead of making the enemy 'Mech walk into a hail of gunfire, they’re more walking through a sandstorm - if it goes on long enough, they will erode, but it’s better to focus your damage.
As a rule, in BATTLETECH you will always be out-numbered and the way you win is by proceeding cautiously, methodically. Every turn in combat should be an exercise in evaluating “what enemy piece can I remove from the board while taking a minimum amount of risk.” Like Austin said, this often means finding enemy 'Mechs or Combat Vehicles with high threat-to-durability ratios. The SRM Carrier/LRM Carrier vehicle, for example, is a nightmare but has less than 200 armor total so it should always be a high-priority target. The Jaegermech is typically equipped with less than 600 armor points on a platform with 4 autocannons, making it a glass cannon. With practice you’ll learn to evaluate these threads instinctively.
As a starter hint, your highest priority in any given turn should probably be to remove any AC/20s from the enemy.
If there’s no exceptional threat on the board in a given turn your priority should be to attack the weakest available enemy with whatever is available while managing your evasion/cover/armor facing to your best advantage.