Certain parts of the ship are awesome to invest in - particularly the ones which can speed up work on 'Mechs. But getting to the Beta Pod and doing Hydroponics will save you some headaches later. (I promise, it's worth it for random events.)
Regarding trouble hitting targets? Remember it's often better to have many smaller weapons which can fire repeatedly than a big weapon which can fire and needs cooling. Or a lot of weapons which will require heat management. Because faster 'Mechs are going to be harder to hit due to evasion... but when in doubt, melee can be a good friend - a good slap can sometimes make something "unsteady" and strip all evasion.
For hunting down those pesky Lights/Mediums which are fast and can build evasion, the JR7-D Jenner is an excellent hunter. (Once you strip the missile launcher out, armor up, and add some sinks.) If you can find it, the ASN-101 Assassin has the Intercept System installed - this makes it FAR easier to snipe enemies. Trouble is, it too is going to need some love from Yang's crew. Once you stop seeing many fast and light 'Mechs... you might want to invest in maneuverable units which can pack a punch. I'm partial to the GRF-1S Griffin, PHX-1 Phoenix Hawk, and ENF-4R Enforcer. Once you can get yourself a Thunderbolt of any kind, you're in good shape to punch over your weight - those things can be made to be monsters and are quite flexible in Yang's care.
Then again, that's just been my experience. The campaign, and career, can be beaten with just about anything as long as you're cautious and know how to work the OpFor AI. Why, there's even some fool who does the whole experience with AC/2s just to prove a point...