First thing that helps a lot is the units you get at the beginning based on the questions at the start. Otherwise, training up soldiers (which is a pain in the ass unless you know how to exploit it) and getting a few strong monster classes early on help out a lot. I usually end up getting a Wyrm, Hellhound, and Golem (I only get a golem if I have a Doll Master leading the unit though, they're too sucky otherwise).
For soldier promotion...well here's some quick tips. Because later on down the line you're going to be primarily using human classes. Anyways, for each individual soldier you get 2 promotion points when you end a battle with a win. So, each full soldier unit (a full soldier unit counts as 3 soldiers) gives you 6 points, so with a squad that has four full soldier units, you can get 24 promotion points per won battle. Since it takes 100 points to promote a soldier, it takes at least 5 battles.
The hard part is getting wins with such a weak unit. The great part though, is sometimes you come across units that are primarily cleric, and you can spam battles against them and get easy wins. I remember there's one exploitable battle in Scene 2 (the first Volmus Mine mission), where you can fight a squad charging from the north that has two clerics and a sole fighter that's in back row. So they only get one attack per round, but can heal up all the damage you do. So you just get in repeated fights with them, rank up your soldiers into fighters/amazons (I suggest having one male and one female leader, each with a full four soldier units in their squad for easy promotion purposes). Then each time someone ranks up from soldier, you make them retreat back into headquarters, switch out the fighter/amazon with another soldier unit, and send them back into battle to rank up another one. lol
It's a little time consuming, but trust me. It saves you sooooo much time and effort later on if you can amass a lot of human characters early.