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I wouldn't consider myself a veteran by any means, since I'm in December myself, but here are the things I kicked myself for not doing or generally would advise someone do:
The big thing is, try to spread out contributing actions evenly among your entire roster. For instance, if you have the CP and there are two enemies near the area you need to capture in order to win, and Cosette, Helmut and Marion haven't done anything in that particular fight; don't just bumrush the base with someone already out - have Cosette and Marion kill the enemies and Helmut capture the base.
People who rank first and second on the list of contributions will get more credit bonuses, but if there are fifteen enemies, assigning your favourite character fifteen out of fifteen kills will quickly ensure nobody else will be eligible to promote (though contributions span further than kills), forcing you to grind everyone else up later on. Bear in mind that your tank/APC when you get it will not earn credits, so don't give them the lion's share of the kills either.
Unfortunately compounding this problem is the fact that credits are randomly assigned; so an Anisette who needs nothing but a Certificate to promote to Trooper Veteran could get fifteen out of sixteen kills and receive only basic-level Attk credits, and Avan could get the last kill and contribute nothing else and then get nothing but Certificates.
Using people in battle a certain amount of times will allow you to view additional fire emblem-esque conversations about their character (as anime tropey as they may be) and eventually unlock a mission which upon completion will either fix a bad third personal potential or give them a fourth one.
Go for enemy Aces. They appear in February(?) and there's usually one in every mission past then with the exception of the free missions, main story missions and most classmate missions. They usually give you plans for weaponry upgrades as well as extra bits for your vehicle.
However, if you repeat the missions again the Aces remain, and their drops become somewhat random - it'll either be a bunch of materials, or one of two possible pieces of enemy equipment depending on the Ace - and you can get some even choicer and well into game-breaking territory equipment here. The worst example I've come across, however, is one of the early captured enemy anti-armor tank turrets which completely outclasses all turrets you can develop until well into late-game; it pretty much one-shots everything under medium tank durability with far greater accuracy than your regular turret, as well as most human targets, and that's not even going into the gatling machinegun that comes with it - again, far greater accuracy than your one can offer, plus far greater damage, a huge effective interception range, and more shots per gatling burst.
I'd advise against getting these things unless you're really struggling; since they completely steamroll the game's difficulty into the ground if gotten early - though in fairness, it's mainly the tank turrets that are game-breaking.
If you die around here, you'll burn so hard there won't even be ashes left. If I want your corpse, I'd better take ya down here!