If you could give a budget that would help...
For an amp I would recommend a second hand amp from ebay, £200 doesn't get you a lot new but second hand you can get something pretty damn awesome.
I don't know that many good amps but arcam alpha 8 (£100ish) (alpha 9, alpha 10) £200ish would be good. Also used nad's are a good bet, something like a c350/370/353/372. Rotel also but I can't give specifics to be honest because I don't know.
For the computer the cheapest way is just to run a 3.5mm cable to rca from your laptop to the amp, but it won't sound amazing. You can get a usb DAC to improve sound. Would need a budget to advise further.
As for speaker cable I would just get some thick copper stuff from qed or some other brand fairly cheap. Don't spend too much, it will make little difference.
Then to connect your tuntables/dac to the amp you need RCA to RCA cable, I wouldn't pay too much, less than £20 per cable. (the speakers are only £100 and they are the most important part, so it makes no sense to spend £80 on a cable).
Good luck, I will try to help further if you give more information and specifics like the type of sound you are after (bright/neutral/warm), but there are people here who can offer much better advice than I ever could.