Yes. You just buy different dies for each caliber you want to load. Any manufacturer's dies will fit (RCBS, Lyman, REdding, Dillon, etc.) on your press, but I really do like Lee dies ad use them most of the time.
The Anniversary Press is (IMHO) a good way to get started. The Lee book that comes with it is very helpful , once you get past the ongoing advertisement for Lee products inside. it explains how reloading works, why each step is done and has a pretty good section of load data in the back.
Once you figure out the powder measure calibration, it works pretty well with most powders. The scale works once you get used to it, but it gets knocked off of "zero" pretty easily, so put it somewhere stable and don't move it.
However, I highly recommend that you also get the FREE load data that each powder manufacturer provides. When you compare the data for the same cartridge from different sources, you may notice that the figures given are not always the same.
As said before, lots of knowledgable people here to help you if you ask.
Enjoy!