I'm a part-time indie (evening/weekend warrior) and I've hired a few artists. Sometimes it went well and sometimes it didn't. Here are some things I learned:
A few people covered this already, but do NOT pay a large amount of money upfront. You'll get fucked. Break your payments into small chunks. I did weekly payments and this worked pretty well. Preferably you'll do your payments at the very end, after all revisions, etc. Some artists are a bit more cautious and demand upfront payments. If you run into this scenario, compromise by paying some upfront and the rest after they finish their revisions. Be aware that the moment you give them the final payment, they lose any incentive to work hard. This happened to me with one of the artists I hired - after I gave my final payment, he did an absolute shit job on the work for that week (for example, I asked him to animate an enemy attack sprite and all he did was use the paint bucket tool to color the character red on every other frame) and made a bunch of excuses when I asked him to make revisions.
Make it very clear from the get-go that the artist should periodically send you progress pictures for all assets that he's working on, especially right after the initial concept phase when he is sketching it out and figuring out what it'll look like. This will help you avoid situations where the artist messages you out of the blue with a completed and fully detailed asset that for whatever reason just looks completely wrong to you. In these cases, you'll have to ask them to completely start over, and that gets awkward because they'll have spent lots of time on it and might ask for more money, etc. If they send you progress pics, then you can completely prevent this, and if they agreed to send you progress pics and don't, then you can be justified in telling them to start over since they didn't honor their end of the agreement.
Make sure you agree on a policy on revisions, possibly even picking a number of revisions that you're allowed to ask for without having to pay more. Similar to my point above, this helps you avoid awkward scenarios where you ask the artist to do work that they might not have thought they agreed to.
Don't put extremely skilled artists on a pedestal if they're uncooperative, don't communicate frequently, and are generally just assholes. This is kind of tough because sometimes you'll see an artist's portfolio and think about how amazing it would be if that person put his art in your game. The issue is that as you get more attached to your game, you'll realize that you have a very specific look in mind for your assets. Oftentimes an artist will give you something that is very well executed but it just won't be what you're looking for. And yes, this can and will happen even with artists whose styles you love. In these cases, it's critical that the artist is someone who can understand your feedback and, more importantly, can respond to it without being a little bitch.