That is a really small budget. To be honest, I think the best suggestion I could give you, would be to keep an eye out for wherever someone is selling used equipment locally, craigslist or maybe facebook. Ask around for friends, try the local wrecking yards. If you don't know how to install equipment, you can not afford what most shops would charge to do anything. Some places have $1/free install, you might be able to find a small amp, or maybe even get a good deal on a used Subwoofer with enclosure or other equipment. Holidays are often good times to find deals, upcoming is St.patricks day in 3 weeks, Easter in 6 weeks, and July 4th will be a big sale day.
I do think your best chance though is finding someone getting rid of an entire system, (amp, speakers, wiring, maybe subwoofer with enclosure, maybe head unit), and just ask them if they will help you install it, (wont hurt). Don't expect to talk someone down if you are needing there help to install, when most shops charge almost mechanic rates.
Sounds like you need a amp. A 2 channel amp would power your speakers, a 3 channel amp would power speakers and a sub. A subwoofer and enclosure with wiring, your budget suggests looking for used.
In the meantime:
I would find out if your factory head unit has outputs so you can add an amp, or if you need to upgrade your head unit. Also start researching about how to install. Its really not that difficult.
If you feel comfortable installing it, or have a friend who can help, and can come up just a little. Speakers $40$-75.... amp $60-$100.... subwoofer $50-$100, (box another $25-$40, wiring $20).... head unit $50-$75.... These would be for NEW prices, but about the cheapest of stuff I could think of. You can find a little cheaper, but this is already going to be really basic..
Just ran across this. This place usually has some good sales.
https://www.frys.com/ads/page106
