I'm on level four, I think, the first one where you can build vehicles, and the one where you have to pick five squad members to accompany you off the top.
I'm finding the game incredibly difficult, in a good way: a nice counterpoint to all the "amass and smash" RTSes I've played over the years.
I keep getting WHUPPED by the Russians on this level, though: when the time comes to break north, those crazy turrets make mincemeat out of my dudes, and the Russians have a few tanks to throw at me as well.
So a couple of questions:
My tank can't be refueled, right? There's no oil in my depot and nobody seems to react when I try the push/repair commands.
And do I really need to push like crazy when told to? My only decent survival strategies to date involve going slow and steady, making progress on my belly by inches. With three vehicles and the other guys all converted to Soldiers, I still get eviscerated, but maybe I'm taking the game "story" too seriously and trying to rush too fast 'cause I'm told to.

Shoot, I don't remember this particular mission. Is it the one where you build your first solar panel vehicle?
Make sure you pick at least a mechanic or a unit with high soldier and high mechanic skill, so you can switch him back and forth by entering the vehicle workshop. Mechanics are far better at driving and attacking on vehicles.
If you're using oil powered vehicles, you can refuel them by sending them to the oil deposit tower you must have built in order to have oil or to your main depot if you have oil stored there. You can also tell an engineer to pick up oil from the depot and carry it over to your empty vehicle. If you're using solar powered vehicles, then you just have to wait till their batteries recharge.
Some turrets are better against vehicles than they are against people. Make sure you're using the right attacking unit. Click on the turret and you can tell what unit they can damage the most (when selected, the little window shows their unit damage on the bottom). Sometimes, you can take your bunch of soldiers, have them crawl and hide behind trees and fire at the turrets with the assistance of a scientist to heal them up.
REMEMBER TO STEAL ANY CAR YOU CAN! Mechanics can fix enemy burning vehicles and then take them for your own.
Glad you're enjoying the game. I'm one of the forumers who recommended it to you on the other thread and I wondered if you were enjoying it.

Some good general points there from ElCAZ - I do remember this mission (sorta) so a few pointers from when I did it (as well as some general advice)
1) Keep an eyeout for the crates appearing - if you hear the sound always try to send out a few men to scout around near your base to find them. This is important in the early stages of the game as it lets you get your resources in - later on you will get access to a research lab and an upgrade to detect (on the minimap) where the drops are - of course its a research you want ot take as soon as you can once you can
2) As El said mechanics are your vehicle users - don't be afraid to multitask troops either - if they are all fighting on foot then set them all a solidiers (with a medic or two as well of course) - if driving set as mechanics (vehicle fighting is based of the mechanics skill not the soldiering one).
3) When you get a lull in the fighting and you have damaged vehicles tell the troops to exit and repair (availible if the trooper is a mechanic). You need to repair as soon as you can.
4) Fallback - always keep an eye on troop and vehicle health, the ai will often focus its attacks on one or two targets, so when their health gets to yellow pull them back from the fight. The ai will often just select a new closest target and go for them instead - giving medics/mechanics time to patch up and send you troops back into the fight. This goes for defending a point (Base) as well as attack.
4b) Also don't be affraid to attack buildings or turrets and pullback troops to heal and then hit again - often you can take out heavy static defences if you make a slow staggered attack pattern like that - though on later missions remember that the ai will often have troops on the map to move to attack you - and they will follow as you pull back.
As for mission 4 as far as I recall you don't have to rush north as soon as your told - instead bide your time and build as many solar powered tanks as you can. Then when you have enough to mount all your troops in vehicles (I think you can get enough resource to do this - if not then you have to risk naked men on the field as well) Move into hte middle of the map and wait.
This is important as the waiting is to recharge your solar cars - they have infinite fuel, but they have short battery lives so they drain out quickly. Then instead of trying to fight your way through simply rush at and through the defensive line to the mission end part. Try to send the main character last so that when he arrives all the other troops are there too (if he gets there early you will be asked if you wish to leave with only the currently arrived troops or wait for the rest - if you wait you will have to make the main character leave and reenter the designated escape zone to triggor leaving again).

Is this the mission where you rush past Russian defenses to safety? I just built a solar buggy for each person and sent them all at once, never lost a guy with that method yet. Shame the other guys going the other direction don't have near as good of luck.

I played that mission a few days ago.
Go to the left of the map and move north.
You come across 2 or 3 Russian soldiers which you can kill easily as you outnumber them.
Move north further.
You can ignore the turret as the people on foot can easily pass it outside it's reach.
The solar vehicle is only in the turret's range very briefly, so there's no danger from that either.
If you wish you can then destroy the building on the east, although some Russian soldiers might rush in.
You should be able to easily beat them.