1. The cartridge is the bit that the stylus plugs into, which hangs down from the tonearm.
2. Good question - I usually leave it too late, I'd suspect. I care about my LPs, but I'm not obsessive. If you're buying 2nd hand, then get a stylus anyway, unless the seller explicitly states that it's brand new. It's worth finding out in advance how much the stylus is for the record player you're intending to buy, as some can be expensive.
Most modern amps don't have Phono inputs, so you might need to buy a pre-amp if this is the case - something worth thinking about.
I'd recommend a
Project Debut III - it's not cheap, but it's an amazing sounding deck for the money - and comes with a built in pre-amp so you can plug it straight into line-in. In other words, it doesn't need either a phono input on your amp, or a separate pre-amp.
The Project also comes with an Ortofon cartidge, which is great quality, and there is a USB version, so you can rip your vinyl on your PC.
I reckon it's a bargain - but there are other USB turntables you could purchase if you wanted to go down this route
