First, go to
http://www.shootata.com/default.aspx
and find a trap club near you under "Clubs and Shoots."
Visit and get acquainted.
You will get a lot of help and advice so you can make an informed decision about what trap gun to buy. My first dedicated trap gun was a Remington 1100 TA and I still have it, even though I have shot many others and owned a fair number.
For .22 shooting, go to
http://www.nrahq.org/compete/
Look under Getting Started, Rifle Programs, and Silhouette Programs.
Somewhere in there you should be able to find a contact to get to a smallbore range.
Go to
http://www.odcmp.com/
Look under Competitions and CMP Rimfire Games
There are a couple of rimfire benchrest sanctioning organizations, too; but I don't have their contact information handy.
Check out my friend's board at:
http://rimfireshooting.com/index.php?
You can get some discussion of most forms of smallbore there.
A custom gun?
You are getting ahead of yourself.
You can try most forms of shooting with a stock gun.
Once you have picked your game, you can specialize. There are a lot of nice factory target rifles for NRA Conventional (bullseye) matches, not much to do but pick the prone or 3 position model.
Metallic silhouette is a distinct event with its own rules and guns.
CMP Rimfire Sporter is essentially a stock gun match, not much to be done customizing beyond a crisp 3 lb trigger pull on a 7.5 lb rifle.
Benchrest is where it gets wild in .22s. The open gun classes are very custom... and very expensive. Tony QCIS on Rimfireshooting.com is a sponsored shooter and can point you in the right direction.