I'm basically still a beginner (about 18 kyu), and in listening to commentaries by dan level players I realize how much more aware of the points they are during the game than I am, and how that enables them to make clear decisions to give something up in exchange for something else. Mostly I (and it seems like most players around my level) seem to be just trying to keep everything, except maybe the sacrifice of a stone here or there. I realize part of this is because I'm not really counting my territory; I'm just generally judging it. So I'm assuming it's helpful to be a little more accurate in counting territory during a game (right?) so my main question is, how did you go about getting better at that? Is it something that naturally comes, or is it worth it to be more intentional about acquiring that skill?

You need to make a deliberate effort. But take it one step at a time. Learn to get a general idea of how much common corner positions are worth. Learn the value of simple endgame moves. Then take it from there.

My counting lesson are aims at people a little stronger than you, but they may still help: http://senseis.xmp.net/?SteveFawthrop%2FCounting

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

A lot of people, myself included at 3 kyu, don't have any particular counting skills, so I wouldn't count on it coming naturally. A method I've heard a lot is count pairs of definite territory, with a captured stone on the board being one pair, and make a reasonable guess at the unfinished intersections. I still do the ridiculous method of one by one counting. Slow, but it works.

If you can, remember a tally for each individual territory, that way you only need to recount each one as it changes instead of the entire board.

PeterN

At 18k, spend your time on easier fundamentals (such as choosing the bigger space or recognising important moves), reading and basic life and death understanding. Start territorial positional judgement when you are about 13k. Then read

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What you can start doing now is counting the score at the end of a game. If you replay pro-games, it's a good idea to count and compare with the given result when you finished.

That way, when you will feel the need to estimate the score during a game, you will already be able to do so reasonably fast. For being able to determine exactly how much territory you can expect from a unresolved position, I think experience is the best guide.