I started many years ago as a kid picking up spent brass at the range. It just grew from there over time. All responses above are great advice. You can also attend your local guns shows, ranges, small gun shops, other internet firearms forums, etc.
True collectors collect "live" ammunition for their collections, and it's primarily what you'll see at the International Ammunition Assoc. trade shows mentioned above in the link. I simply collect empty cases... it's easier, less expensive, mailing efforts are not regulated/hasseled, it's just safer, etc.
Collecting the cases has also assisted me in making decisions as to which next building project I've wanted to pursue or eliminate. It's amazing at the number of wildcats that have been developed over the years. I believe PTG has over 35,000 reamer prints, and clearly, that's not nearly everything out there.
I've been very fortunate with my collecting. I've met many great fellow shooters in person and through communication efforts from all around the world. Send me your mailing info in a pm, and I'll pay it forward to you with a box of some duplicates I have if you're serious about getting started.