Speed vs. accuracy
IMHO, you can pick one extreme, speed, or the other, accuracy, in reloading equipment. If you want some of each you'll have to compromise.
Since you say accuracy is more important you should (again, my opinion, and not all will agree!) go with a single-stage press, where you can carefully control all the variables for each round. Unless you're shooting a lot more .308 than most of us, this won't take too much of your time, and you will be turning out a more accurate, repeatable, round.
You want a heavy, stable press. Cast iron or steel. The RCBS Rockchucker is probably the most common of these, and is the "gold standard," but there are several others, for about the same $$, that will do just as well. There are also even heavier models that sell for more. Any nice heavy sturdy single-stage press that you can get a deal on will do fine for .308's, and any similar round.
(If. later in your reloading career, you decide to go progressive, you'll still want a good single-stage for load development and such. So it's never a wasted purchase.)
For greatest accuracy, you will also be doing putzy things like weighing each and every powder charge, and hand-seating the primers. All of this takes time, but allows more control over the quality of the end product.
Exactly how much time and effort you want to put into each round, before you put that round into your weapon, you have to decide for yourself.
I guess it boils down to what you call accurate. If minute-of-deer accuracy is fine, then go for that, and be happy. If punching one hole in a paper, at 100 yd. or more, with 10 shots, is your idea of accuracy, well, that is another matter altogether.