A good set of speakers compliments a nice DJ setup like the right glass of wine compliments a nice meal. Sacrificing too much quality on either end will leave the other longing for more. That being said thinking about your budget and investing for both sound and a setup will leave you happier in the long run.
If you are starting out on a limited budget and just want to learn how to DJ and you already own a home stereo system or even a nice pair of computer speakers then by all means use em! When it comes down to it, it is more important to be able to DJ than have good speakers with nothing to play through them. The only recommendation would be to make sure you have the outputs and input connection compatibility you need, and make sure you plan to improve your sound down the line!
As technology improves and the prices of studio monitors become more and more reasonable, the line between a home studio and a real studio slowly vanishes. It is very possible to invest very modestly and still end up with great, studio quality sound.
Normally a good pair of active studio monitors (active meaning built in amplifying, not needing an external amp) will be the best choice for someone looking to outfit their home, studio, or home studio with high quality sound. The price and quality of studio monitors ranges quite a bit and you should be able to find a pair that fits your budget no matter what it is.
This doesn’t mean you are able to go snag up a pair of used monitors for $100 and slap them in the corner of your basement and call it good, but for a few hundred dollars and some attention to things like acoustics and monitor positioning you could transform a modest space into a full fledged home studio.
A good set of cans won’t directly effect the sound coming out of your speakers. That being said, a high quality set of headphones will allow YOU, the DJ, to hear the mix better. Nice headphones do a couple of things, first they block out the external sound coming in. Second they provide accurate highs, mids, and lows which you will use when cueing up your next track.
Basically, good headphones will allow you to hear the details you need to help you mix while a crappy set of headphones won’t.
So in essence what a good set of headphones will do is indirectly increase the quality of your sound because your mix will sound tighter. Check out the DJ Headphones Section of our DJ Gear & Software Section for more information on headphones.