Unfortunately, there aren't too many ways to cheat the learning curve. "Write a lot" really is the first-order answer, and it's usually the one given by professionals that don't stand to make any money from their answer. Writing workshops and other 'accelerants' seem to be very questionable.

A friend who was recently signed as a science fiction author at Tor suggests that you could try compressing things that you have already written. Take something like a blog post, and express the same ideas and content using half the words. This will help with clarity and concision, as you get better at the exercise- one of the telltale signs of an amateur writer is that they use words inefficiently.

Things that aren't typically suggested that worked for me and may generalize:

I. Read mediocre writing. If you only read good writing, you can't tell what makes it good; it just looks like all writing is good. By comparing good writing to mediocre (or bad) writing, you can see what the good writers did that the mediocre writers didn't (or, as is often the case, what the mediocre writers did that the good writers didn't). My writing only started improving after getting out of English class and replacing cultured reading with mediocre fanfiction.

You want to spend time in the position of the reader, thinking to yourself "I wish that the author had written this part differently," so when you're doing your own writing, you notice when you do the same mediocre thing, you change it.

Good, professional writing is what you seek to emulate, so reading it is valuable. However, it covers up the drafting process, which is how you actually get there. This is somewhat analogous to math, where you want your proofs to look like the ones in the textbook, but the way you generate such proofs is much messier.

(EDIT: As has been discussed below, reading mediocre writing is, by itself, probably a bad idea. The value comes when you deliberately train yourself to recognize bad things and how to make them good.)

II. Spend time in writing communities. I've gleaned a lot of value by reading a blog that reviews MLP fanfic, mostly from getting three doses of "this writer did this and this, but it didn't work for these reasons" a week. More recently, I joined a proofreading team, where one of my co-betas yelled at the rest of us about hyphens, en dashes, and em dashes, resulting in me finally using them correctly (and learning how to type them, which depends on your system).

III. SICP. Computer programs are complex. Only reason we can write them is that we've developed some sophisticated complexity-controlling techniques, which SICP covers. These techniques generalize to other areas, like math, physics, and, most importantly, writing. I spent years unable to write anything coherent longer than 500 words until Dr. Abelson started talking to me about wishful thinking and black-box abstraction.

IV. Learn to type. You will (1) type faster, (2) use better technique, thereby reducing injury potential, (3) hit the wrong key less often, reducing the base rate of typos and unnecessary keystrokes, (4) be able to look at the screen, which means you (a) will notice more of the typos that get through, (b) can see what you've written, and (c) have better ergonomics, and (5) you'll achieve automaticity, which will free up cognitive resources for other things. This, I think, is another area where programmers are ahead of writers, even though they should be doing essentially the same thing.

Standard recommendations that bear repeating: read Strunk and White and, if you don't have any writing of your own to work on, beta read somebody else's writing.