Don't forget you need a hammer too!
OP, I love reloading. It's one of my hobbies.
I recommend starting with just one caliber, probably the most expensive one you shoot a lot. As has been previously posted, you don't need a lot of space, depending in your equipment. The Lee hand press is a great recommendation, as is the handloading kit. The handpress will allow you to pump rounds out slightly faster, whereas the handloading kit is probably the best value. But you will need something sturdy to pound on with a hammer. A work bench would be best but it doesn't sound like its available for you right now.
Definitely get several reloading manuals. You can find used ones on eBay or amazon. I bet there's even a few people here who don't need their older ones anymore. The older ones are still valid, for the most part. I recommend buying your components based off of whatever book you have. For example, don't buy a certain type of powder if your book doesn't have listed recipes for that powder in that caliber. Of course you can get all these recipes from the powder manufacturers website, but I personally prefer the books.
You will definitely invest a lot of time into it, but I think you will probably find it rewarding in the end.
Just remember - you may not save any money on ammo. You will just shoot more with the same budget you had before! Or not. But that's certainly been my experience.