As axefrog mentions, UDK is the older product and unless you've already started a project with it you would probably be better off with Unreal Engine 4. That said UDK is still available and will still work if that's what you prefer to use -- I've never tried it myself and can't comment on it however.

I will comment on Unreal Engine 4 and say that it's a pretty amazing deal for hobbyist and small indie developers. You pay $20 monthly for each team member that will be using the engine and then just 5% royalties, and that gets you the full Unreal Engine including source-code access! It's also worth noting that you don't have to keep paying your $20 each month if money is a concern; by paying the monthly subscription you get regular updates, but you can instead choose to pay your $20 once, download the latest version of the engine and source code, and then cancel your subscription -- you won't get updates, but your copy of the engine will continue to work and you can resubscribe at any time if updates become important. Note that you of course still have to pay your 5% royalties once a product is released.

This is a professional engine. It's proven, it's very capable, and it can be used to create a wide variety of different types of games including 2d games if you wish to do so. There are also plenty of learning materials available.

You might consider Unity or Godot (free and open sourced, but less mature and with less resources available at this time) as possible alternatives.

That being said, all of these engines (perhaps less so when considering the less mature Godot) are large products with powerful editors that are often best suited to teams of skilled developers; they certainly can be used by a solo developer as well, and are now priced to make that an attractive option, but for some people they're overkill and something simpler might be more suitable...

What sort of games do you actually want to make, and how experienced are you? We may be able to recommend other options that might be better suited for your usage.

Hope that helps! :)

Unreal may well be a good option for you in that case -- it was originally developed for first person shooters and is therefore very well suited for that purpose. Unfortunately you can't try it out for free, but at just $20 for access to a world-class engine I don't think it's very affordable to give it a try.

If you're looking for alternatives you might also consider C4 Engine, which is a little more expensive up-front (licencing info) but doesn't charge royalties.

Does Unreal 4 really require a quad core to run? I know that previous versions of Unreal were able to run on just about anything.

Honestly I'm not sure, but that is what they list in the system requirements in their faq. I'm afraid I don't have any systems with lower specs that I could test it on for you to see if that's actually a hard requirement or not. Note that those system requirements are for developing with the engine and depending on the game you develop don't necessarily represent what players of your game will need. For comparison, here are the minimum and recommended specs for UE3/UDK.

This and this may be good starting points to really answer your question, but you'll have to look through them for yourself.

Hope that's helpful! :)

Does Unreal 4 really require a quad core to run? I know that previous versions of Unreal were able to run on just about anything.

Games built using Unreal 4 don't necessarily need such high specs, but the editor itself is kind of a beast. At the very least you are going to want a pretty high-spec graphics card.

I ran it on a 1st-gen Retina MacBook Pro (quad-core i7 + 16 GB RAM + SSD) for a short while, and had to switch back to running it on my Windows desktop instead - the MacBook was overheating and beachballing on a pretty regular basis, I think mostly due to the NVidia mobile GPU not keeping up.