The standard line with reloading is that you won't save any money, you'll just get to shoot a lot more for the money you spend. Reloading is basically a tradeoff between time and money. You can buy a single stage press for very little money, but then you have to perform each step of the reloading process individually. First you resize all your cases, then you prime them all, then you measure powder into each case, then you press a bullet into each case, and finally you crimp each case. This is a good way to learn about the process of reloading, but it's relatively slow. For each step, you have to reconfigure the press. Reloading 100 rounds could take hours.
The other type of press is a progressive press. Check out dillonprecision.com for the best ones. This is a bigger (and more expensive) machine that holds all of your dies and performs all of the reloading steps simultaneously. You just insert cases into the press and pull the handle, and it sizes, primes, measures, loads, and crimps them in a round robin fashion. With this type of press, you can crank out 1000 rounds in a couple of hours.
Some people enjoy the methodical activity of reloading and use a single stage press. You can get started with everything you need, including a single stage press, for as little as $100. I personally just want to get the reloading done as quickly as possible, so I use a progressive. For a progressive press, expect to pay $500-600 for everything you need to get started. There are basic kits to get you going, but there are also lots of little accessories you will need.
You also have to consider where you are going to do this. You can reload in a small apartment, but I suggest you set things up so that you can move the press if need be. I have a Dillon progressive press that I bolted to a piece of heavy plank about 1x3 feet, which I then clamp to a heavy desk. It's a solid setup, but I can take the whole thing down and store it on a closet shelf.
You also have to consider dirt and lead. The process of loading isn't bad, but cleaning cases in a tumbler is a real dirty business that you should best do outside the house. My apartment has a utility shed where I keep the tumbler. Also, if you have kids around, you should be concerned about the lead released into the air during the depriming process. I'd better capitalize that. DO NOT RELOAD IN A ROOM WHERE KIDS HANG OUT.
Before you begin, follow the advice of the others here and get yourself a good reloading manual (such as Lyman) and read up about the process. Safety is paramount. Reloading is not something to do casually. Don't expect to turn on the TV, put your feet up, and casually reload a few hundred rounds while relaxing at night. You have to concentrate on the process and be vigilant at all times, or you could end up with a double charge of powder that blows up your gun, or a squib load that leaves a bullet lodged in the barrel.
