First IMO based on your strict budget ($350), I can't imagine any scenario where a quality audiophile system could be assembled, not even a decent mid-fi quality system. I don't know enough about low-fi offerings to make a recommendation. My recommendations would be 1) start saving your money. 2) determine what will be the source of your music (cd, dvd, vinyl) 3) find a quality hi-fi shop that allows you to sit down in a "listening room" and start listening to equipment. A quality shop will be able to set up the equipment without any extraneous (read switchers) components in the path. They should be happy to pick up, move around, switch out equipment mixing and matching source, control, and loudspeakers until you find the combination that moves you. The shop may also provide demo units and perhaps consignment pieces that would allow you to save a few coins.
Regarding the allocation of your funds, my recommendation would be the complete opposite of that provided by BoxterCharlie. My belief, and approach in my own system, is to put the empahasis on your source hardware (and software), then your control (amp/pre-amp or integrated) and then your loudspeakers. The logic is that if your system is producing sub-par music, that is what your loudspeakers will emit. The better the speaker, usually equates to a more revealing speaker which will certainly reveal all the warts of the equipment upstream. So sacrifice initially a little bit on the speakers in order to get the best music reproduction equipment possible.
Regarding the allocation of your funds, my recommendation would be the complete opposite of that provided by BoxterCharlie. My belief, and approach in my own system, is to put the empahasis on your source hardware (and software), then your control (amp/pre-amp or integrated) and then your loudspeakers. The logic is that if your system is producing sub-par music, that is what your loudspeakers will emit. The better the speaker, usually equates to a more revealing speaker which will certainly reveal all the warts of the equipment upstream. So sacrifice initially a little bit on the speakers in order to get the best music reproduction equipment possible.