stuff you'll need -
microphone, xlr cable, mic stand, audio interface, usb cable, computer, recording software, headphones, amplifier, speakers, perseverance...
If you've already got some of these things them use them - a stage mic like an SM58 may not be ideal, but it'll work and better than nothing. Otherwise the Audio Technica AT2020 is probably the best of the low budget mics..
Audio interface - I got a cheap Alesis off ebay, but there can be compatibility issues with older interfaces and newer operating systems and drivers may not work with Windows 7, 8.10 -
ASIO4All often works
As far as what's available new - Behringer are the cheapest - their stuff ain't always the best, but it'll do, other than that Steinberg, Alesis, M Audio, PreSonus, Focisrite, Lexicon and a few others do a range of products in the sub £100 area - generally inpits for XLR and 1/ inch jacks, phantom for a condenser mic, latency free monitoring, a midi socket if yo have midi gear and level controls for inpits and outputs are the kind of functions you should look for - also it's handy having a range of outpit sockets - phono outs come in useful if you're using a hifi amp for monitoring
I'm not familiar with Apple Macs, so can't comment on that area
Software - most audio interfaces come with some free recording software - Cubase is my preferred choice, but that's because it's what I learnt on, so I'm used to it - whatever you end up with there will be a steep learning curve and lots of cursing and googling and watching youtube tutorial videos..
Headphones - you'll want these to listen to the backing tracks when you're overdubbing your sax - closed back designs are preferable so you don't get any of the tinny headphone noises spilling into your recordings... having said that I've overdubbed stuff while listening back over speakers and the spillage wasn't noticeable in the final mix and there are many classic albums that have the drums and guitars bleeding over into each others mics but still sound great, so don't get too hung up on getting perfect isolation.. Beyer Dynamic, Sennheiser, AKG, Audio Technica and Sony all make good headphones, get some that are comfortable, you'll be wearing them for hours..
Headphones designed for DJ's may not be very neutral sounding
Speakers - a lot of the cheaper speakers sold as studio monitors aren't worth bothering with, a good pair of hi fi speakers from a reputable company like Kef, B&W, Castle, Mordant Short etc will be quite fine for home use. I've been mixing and mastering for clients using my old Kef's for years and they work for me because I'm used to them and know what every album in my collection sounds like through them. I wouldn't touch Yamaha NS-10's with someone else's bargepole, I just end up trying to EQ all that ugly midrange.
I have mixed on headphones, but prefer a pair of speakers so I can listen while I'm in the kitchen - and it's less tiring on the ears