Hi Tommy and welcome to the world of recording and synths

I can definitely help you out because I was you in 1990. It is always fun to see someone have that toe in the water and interest to learn. I apologize if this sounds like a 'dad speech' as I definitely don't know it all and am still learning new things all of the time myself. Things have changed a lot even in the last decade. Like you, I taught myself to play keys which is why I got into MIDI and DAWs because you really only have to play it right once lol. Once you have the data ya can clean it up however you see fit or twist it 180 degrees to get something else cool. That is the beauty and sometimes pitfall of the digital realm. Some creative sounds and effects can be found by modifying things digitally but it also becomes a rabbit hole where a song is 5% content and 95% editing lol.
So, your machine is more than capable to handle digital anything most likely. Windows 8 would not be my first choice but if you know the OS inside and out and how to dig around in Settings, go for it. It will help remove a factor of problem solving when trying to figure out the problem of a missing device or no sound etc. You are most likely fully capable of recording sound with the mic or line in on the sound card already present. You will want to mute the mic input if you are not using it as it tends to pick up system noise and throw it into recordings. If you want a more professional experience, I would definitely suggest a USB outboard box like an M-Audio or such to do the DAC and have less system noise. There are a lot of them out there, which is good in a way because you can get older models cheap.
You will also need some sort of MIDI input/output for your computer. Most newer synths use USB but I still prefer the chunky DIN type. It kinda depends on what you want to do musically but judging by your musical tastes (which are similar to mine) this will help your pc chat with older gear. OKAY so screw the $5.00 USB dongle thing you are looking at on Amazon in another tab haha. That thing will frustrate you later as it will not pass MIDI sysex data. You will want a real deal MIDI outboard box. I still use one of these lil guys which does great and is cheap on ebay.
https://www.ebay.com/itm/M-Audio-Midima ... :rk:2:pf:0 Others may have a different opinion or you may just decide to find a box that has audio ins and midi i/o in one box and cut your overall cost somehow.
Okay for synths. Man it is a huuuuge world of those out there with hardware and software synths absolutely coming out of the cracks. People will vary wildly in their opinions on such and I will probably get shat all over for my recommendations lol but here are some tips and a suggestion or two to get ya going. For the old school space synth you will definitely need two things: an arpeggiator and pads

You will also need knobs to run your cutoff and resonance settings that are so important for that sound. You will also want something that is pretty easy to use and doesn't involve tons of menu diving but that doesn't exist lol. I would not recommend getting an analog beast for your first synth. They will be limited to you in usefulness and usually require maintenance and cost a buttload of money. So you will most likely end up with a Digital/Analog hybrid or ROMpler as your first synth. For 80s music, the DX7 ruled but it is a pain to edit without software and will need maintenance and shipping will be the cost of a cheapo synth lol as they are tanks.
For my daily space jammings I use a Yamaha CS1x, a Roland MC505, and a Korg Poly 800 or Microkorg. Lots of arpeggiators and knobs in there lol and they will most likely be found for 200-250 dollars. For your 80s FM sounds, you can go from small to large but I actually enjoyed the early PSS 480 or TQ5 as they were easy to program and cheap. I honestly use a softsynth for FM though when I go to use it.
As for recording software, Audacity is FREE and will handle anything you throw at it but is kinda basic compared to paid apps. It can handle multitrack and editing as most all DAWs can. Reaper is another great option as it steps up the game a bit and does that along with plugins etc. Reason is another great piece of software that is basically made to do our type of synth music. It is the complete package and has great depth of features to control every aspect of softsynth based music production. It is probably the only thing that could kill Fruityloops (goofy name insanely great software). Acid
https://www.magix.com/us/music/acid/ is also great for looped music. I remember when it came out I considered it cheating because I no longer had to sit there with a calculator and make sure loop lengths matched smdh. Ableton is another option but I just never got into it. There are folks that love it but I just didn't enjoy it. Maybe I am old lol. I also still enjoy Rebirth for acid jams. It is a bear to get running on Win7 and up tho. There is a fun site
https://www.audiotool.com/ that gives you lots of goodies in your browser with drag and drop setups, wires, and old BOSS stomp pedals lol. Runs in your browser and is free. Hopefully that will give you some things to look at and think about and for others to yell at lol. Maybe take a look around the internet and do some shopping and let me know what you find in your price range/ask any questions about it. I mainly do repair these days and hardly ever record and just jam out into space

Gotta run for now.