I just finished installing 100% new door trim on all my interior doors and will begin replacing all the baseboards this weekend.
When butting up the baseboard to the door trim, what do you do? Do you just butt the baseboard up against the door trim? Do you cut the end of the baseboard at a different angle (other than 0 degrees)?
Either you are asking an incredibly basic question or I am completely missing the scope of your problem. There would typically be no reason to make a cut other than plumb and square when butting base into casing, unless you have an oddball situation and you did not give us enough information to clarify the problem you are having.
You have a ton of options. If you plan to butt the new baseboard right up to the casing, and the vertical angles do not match, cut the edge of the baseboard to the correct angle (using a bevel gauge) so it fits tight with no gap. In other words, if the door casing is five degrees away from plumb towards the baseboard, you need to make up the difference in the cut of your baseboard. You can't count on 90 degree cuts to fit unless everything is square and plumb.
If the casing is not flush with the wall, you can back-cut the baseboard so only the front edge touches the door casing, or else you can measure that angle too and cut your base accordingly.
OR, what I like to do is make plinth blocks, in which case it doesn't matter if the side casing ends up a little proud of the base or vice versa.
If the two boards are supposed to be the same thickness but they don't quite match up (and you'd like them to!), shim the shallow one so the two boards become flush after you nail them.
Also, how you do it depends on your base cap. If it's a one-piece base, then you need to match all the angles or else scribe one piece to another. If you're using a separate base cap molding on top of the baseboard, you can back cut the base and the cap will cover the material that you removed.
It sounds like your base is thicker than the casing, causing it to be proud of the casing. One option is to return the base you have purchased for thinner base. If that is not possible or no base is thin enough, I would opt to change the casing to thicker casing. Whatever you decide, you will want to settle on a solution that will work for every opening so that you have a consistent look and feel to each joint between base and casing. Plinths are one solution, a backband could also work. If you want a quick and dirty solution you can cut a chamfer on the end of the base to ease the transition to the casing. Shimming the casing off the wall would work unless you need to shim it a large amount. I would personally get thinner base or thicker casing and make it look clean. The last option would be to run from the job, but your wife would not be too pleased with doing that so be careful!