Active monitors, like any engineering solution, have a downside and an upside.
The upside is that the amps are custom-matched to the woofers and tweeters, and the IM distortion generated by passive crossovers is eliminated. The sound turns out very clean and dynamic. Also, you have one less box to place and you don't have to buy speaker cables. You may need quite a long interconnect though.
The downside to actives is that you are limited to the quality of the electronics installed in the speakers--no upgrades are possible, realistically speaking. You have to go with new speakers AND new amplification if you ever want better than what you have.
In your situation I would not choose active monitors unless I got a great deal on a pair whose sound I liked. I would look for a good entry-level integrated amp (perhaps a NAD C320BEE, or the Onix amp you mention) and passive monitors. BTW, an integrated amp is one with both the preamp with its controls and the power amp in the same box. It's a very cost-effective design and can sound as good as separate components.
I would use my computer as a source as long as having it in the same room with the stereo were practical and quiet. I'd get the best USB DAC I could afford, starting with the minimal Griffin iMic or similar and going up through the Digidesign MBox to the Apogee Mini-DAC. (There are others too that are well-liked here--do a search on USB DAC in the Forums.) I'd save up to buy decent cables later, both interconnect and speaker, and for now use something at around $30 for the IC and $2 a foot for the speakers.
