Me and my mate have the occasional jam in my bedroom, me on guitar, me vocals and him on the bass.

I was wondering how I should set the room up to get the best sound?

And also, I use a karaoke machine with the microphone I got with it and it sounds rubbish. What shall I do to improve it? Ive tried EQ'ing it to make it sound a little better, but it still sounds terrible.

Is there anything I should invest in? Like if i bought a cheap PA system, would we plug both of the amps into it aswell or is it best to just use it for the vocals? Would it be worth buying one and would it be much of an upgrade compared to the karaoke machine?

I have an old guitar amp i can use instead of the karaoke machine, but I think that sounds just as bad as it doesn't have an EQ on it.

Any ideas are much appreciated
I'd use the amp you have, and just keep your levels in check. If you're just jamming, you'll be fine. If you end up with a drummer too, then a cheap/used PA might be a good idea.