In addition to what has been said above, and since I didn't see it mentioned.
Early on, with low-level pilots, salvaging one of those Assaults in the clash of titans mission is close to impossible.
Your precision shot accuracy is _horrible_ if you go for anything but a leg/side torso from that side and you can just forget about headshots.
Also, early on most enemies have reduced armor, so going the "make them fall on their face repeatedly until the pilot dies" route is equally impossible (the mech will be cored before that happens).
You _can_ try to leg the target, so you'll get at least two mech-parts but that's about it.
As for general advice, take your time. Since you have done the first campaign mission, you are free to travel - make use of that and get some better mechs (replacing the Spider has already been mentioned)
Mechs to look out for (IMO):
Contrary to Spartakus' advice, when it comes to light mechs, I'm not a big fan of the Firestarter (I guess I qualify for being a heretic
) but much prefer the Panther (LL + 3 ML + max JJ + armor and HS).
Otherwise, as Spartakus said, Jenners (drop the SRM-4 for more armor) and Javelins (the F variant is preferable, IMO) are very much servicable.
For mediums, Centurions, Hunchbacks (I'm not a big fan of the AC/20 one but the Hunchie P (lots of ML) is really good), Shadowhawk-D, Wolverine-K, Griffin-S and Kintaro are all very solid mechs
Once you get to the heavies, avoid the 60 ton, 5/8 mechs like the Dragon or Quickdraw (5/8 refers to the table-top speed of walking/running. Faster mechs need larger engines and have therefore less tonnage remaining for armor and guns) - for the same reason I avoid the 50 ton Trebuchet (5/8) or the 40 ton Assassin (7/11 - holy s**t, there is hardly _any_ tonnage left for stuff) and let's just not mention the Cicada, btw.
Thunderbolts, Catapults, Grasshopper (excellent crit-seeker), Archer (good buffs for missiles), Warhammer (massive buff for energy-weapon damage), Orions, Black Knights and Marauder (ultimate head-capper) are all very good.
Edit: Forgot the Rifleman and the JagerMech, both also quite decent - to be honest, aside from the aforementioned Dragons and Quickdraws, I can't think of a really _bad_ heavy mech.
Some are niche mechs like the LRM-20 Catapult, but all are very much workable.
On a final note:
I'm personally a jump-jet-aholic. Everything I have mounts them and usually the max. number. Yes, it takes tonnage away from guns, heat-sinks and armor, but the tactical mobility and flexibility more than makes up for it, IMO.