You share the public key, the fingerprint is just a way to make sure someone has your real key. The key ID is an easy way to identify your key, but it's not as unique as the fingerprint (which is unique to every key).
Usually uploading your public key to a keyserver is good enough, and probably what the people sharing only their fingerprint do. Then anyone can get your key from a keyserver, and you can just share the fingerprint more widely, it would make a decent signature...
You should probably read general info on Public-key cryptography and PGP.