Yes you have basically got the idea. You have one system that fits on to all your different lenses and uses any/all of your filters. To attach the system to a lens you use a special adaptor ring which needs to be the same thread as your lens, so if you have three lenses of different thread sizes you will need three adaptors, the beauty is that each adaptor is on a few £ comparred to the cost of many filters.
The difference between the A and P system is the size of the filters that you use in it. I can't remember without checking what the sizes actually are but I believe the A to be smaller than the P. I myself use the P system and my largest lens is 77mm thread size, this system works great with all my kit.
I have a few grad ND filters and a couple of ND's as well, this suits for the majority of my needs. Also be aware that you can use other brands of filters as long as they fit the system, most manufacturers offer filters specifically for Cokin systems, usually at a discounted price too.
SRB film hold a massive stock of different filters from different manufactures at very good prices. They do mail order and I have used them several times without any issues.
http://www.srbfilm.co.uk/
Hope that helps?