One of my projects has been improving my writing. Succinctness is one way to do this and is probably the area in which I need the most improvement. Constantly saying "I haven't had the time to make this shorter" is saying "my time is more important than yours." This is bad when writing personally (or in your weblog), and disastrous when writing emails in a business context.
Having said that, I can recommend Stephen King's On Writing as a general resource. I recently purchased and have been working my way through William Zinsser's On Writing Well, which I can also recommend for non-fiction writing.
If free is more your bag, copyblogger.com is full of good tips, including the most recent post, "Are you writing with clarity". Seth Godin is one of the best I've seen at the craft of weblogging. If you want to become a better writer, then one of the best things you can do for yourself is to read. All good authors read other good authors.
As far as weblogs go, some people might argue that they're only writing for personal interest, and therefore they should not have to be terribly concerned with writing well. That is unmitigated bullshit: if we weren't interested in having our thoughts read, we wouldn't go to the bother of setting up weblog software and posting to it. So, if you're interested in a wide(r) audience, it's in your best interests to hone your writing. If you're not interested in getting your points across clearly, then you would be doing everybody a favour by saying "this is just a braindump, I don't expect you to understand it so don't waste your time reading it" at the start of every post.