Welcome to the forum. I'm in a similar situation in that I have a music background. As far as hooking the computer up for music, I have mine hooked up 99% of the time I'm on the computer. Right now I'm listening to classical music from a radio station through my computer as I type. I live in the boonies and could only get most stations through the computer. AOL Radio often. So, it makes good sense.
As to the type of speakers, we all have different ears and only you will be able to tell which speakers fit you best. I use an Onkyo stereo receiver and have been pleased with it, so I think you're in good shape there. So, I think you'll have to try some speakers to get an idea of what you like. You will get a lot of advice for your price range as there are many fine speakers out there. My standard advice for new folks asking me is to try Monitor Audio if you listen to mostly classical and PSB if you listen to mostly rock. I think, overall, PSB's are better, but I chose Monitor Audio because I think they work better for classical. B&W are also on my short list for new folks.
I would offer a word of advice, because you have a music background. Some of our brains listen in a way that like to have the separate instruments mixed together for us. Some, like me, like to hear the parts separated and detailed and our brains are able to mix it. It's my belief that the more background in music you have, the more we like the separated sound and detail of the different instruments. For that, I've heard a very positive difference when listening to 3-way speakers over 2-way speakers. Many of those speakers may be out of your budget. I would encourage you, however, to give a listen to Axiom M-60's. They are Canadian, get rave reviews and are 3-way. I've never heard them, but they seem to fit the bill for what I've always looked for and are in your budget. I would also suggest considering stand-mount speakers and a sub. Again, this combination will give you good separateness and detail. Be careful to get a musical sub instead of a "boomy" sub if music is your goal, as it is mine. REL subs are the best that I've heard, but there are many good ones out there that I've not heard and folks here will suggest these.
Good luck with your search as it is a great hobby for musicians.
John