Bought a guitar second-hand and it's needing adjustments all over: intonation, action at the nut, truss rod adjustment, maybe pickup height adjustment. Rather than take it to a shop and pay whatever they would be asking, I was thinking I could buy all the tools necessary to learn and do it myself from now on.

I watched a series of videos on YouTube and it seems all the tools needed are alan keys, feeler gauges, a scale (small ruler), and nut slotting files. Are there kits out there that include all of these for a decent price or should I just get the tools separately? Should I just take it to the shop?
It would most likely be cheaper to just source the parts individually compared to getting a kit. Sites like Stew Mac sell tools as well, but I find them to be on the pricey side considering what you get. The hex keys, feeler guages etc... all can be purchased at a local hardware store or can be found on ebay/amazon for much cheaper than dedicated guitar places.

You could probably fashion a nut slotting file from different feeler gauge sizes or find a torch tip cleaner with different sized files if you really wanted to save some cash. Personally, I would probably just buy a proper set of nut saws and buy the rest at a hardware store.
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I'm all for hands on learning. You can look for guitar tool kits on ebay or amazon. They have a variety of different kits that come with everything you need.

When it comes to actually using these items. Make sure you understand what you're doing and what can go wrong. Truss rod is always said to be dangerous to play with. It isn't as long as you stick to small turns and know what you're trying to get done. If the guitar doesn't need truss rod adjustments then leave it be. I have done it once or twice to fix some dead notes.

Adjusting the bridge is pretty simple. Truss rod adjustments will probably need to be done when you lower the bridge a bit. It's about what you prefer and what you're trying to play.

Just read up on what you need to do and you can take care of it. Be careful and don't break stuff!
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Two things:

Buy a book called "How to make your electric guitar play great" by Dan Erlewine. Don't rely on YouTube vids; they're definitely not equally useful. All of the information in that book is on the money.

I generally do my own routine setups and adjustments, but Gary Brawer has, for the past several years, done a Great Initial Setup on any guitar that's new to me. In several cases, this has involved a PLEK run, fret superglue and tweaks such as nut filing, saddle shimming, etc. What I get back is a guitar with a flawless setup, and a baseline "known good" setup that we can compare to (and compare with my other guitars) as/if things change over time.
Don't cheap out on nut files or fret leveling kits. Everything else can pretty much be done with basic stuff you get in those large starter packs from chain hardware stores. Anything specialty you just pickup as you go.

As dspellman said, the best step by step stuff comes from a handful of books that are out there. I don't have the one he recommended, but I'm sure it's just as good as any that I've picked up. It's worth it, looks good on your bookshelf, too.
I agree about the Dan Erlewine book. It's a great reference. YouTube videos are OK for some things but they all tend to leave out a lot of basic information. As for tools I also agree that they can get expensive but a few basics like a good fret file, a set of nut files and a few quality basics tools are really worth the investment. Most of the other stuff can be purchased at a hardware store. It's really all about the knowledge. The tools are just that, tools to let you do the work. You need the knowledge first. Good luck.

I have been apprenticing with a great luthier for almost a year now (8-16 hours a month) and many of the things I thought I knew turned out to be completely wrong. I am learning so much about the small details now and love it.
Yes I am guitarded also, nice to meet you.