Hey Charlie!
The folks above are spot on. Yes you will need some others more local to you to help on aspects of this because tracking down stuff is different there than here in some key ways, especially on the used market where for your budget you will get the most for you money without question.
But in the meantime let's see if we can line a few things up.
1. Are you using your computer or similar device as the "source" for the music? You asked initially about computer speakers so I think we are assuming this is the case.
2. How much area do you have available to you and how do you see this stuff being set-up?
3. What kind of music do you mostly listen to?
With the budget you posted, you are a little limited. But the last thing you will want to do is get a bunch of cheap pieces. it will be better to keep things simple initially with just a couple nicer items. Those will probably be a receiver of some sort and some speakers that fit the bill. Those will still be a challenge within your budget but it can be done. I have numerous pieces in my collection well within your budget that would make for an outstanding 2 channel set-up using a computer or phone as a source.
Break this effort into 3 areas:
1. Power - Stereo receiver or separates which I am sure folks on here won't be able to resist telling you about.
2. Speakers - a pair of decent speakers that are of the appropriate size for your situation.
3. Source(s) - Computer is an easy one, as is the phone. You can pick up outstanding used CD players very easily. Turntable gets into another realm you just don't have the budget for currently but you never know what you come across.
The key to this will be versatility and the notion that you always can move up the ladder as you have the funds to do so therefore nothing needs to be considered permanent. Just buy well.
That's all I would throw at you for now.
