Competitions are great. You could also maybe contact someone from this site and send them some beer for tasting.
You generally want to wrap the bottles in a bit of wadded paper and cover with a plastic bag to keep any broken bottles from leaking real bad. Then pack well in a box.
I like to use heavy cardboard tubes that I get from work. They are nearly indestructible and the bottles fit well after wrapping inside a padded envelope from UPS.
Which brings me to my most important point. NEVER use the Postal Service for shipping beer. It's illegal. However, if you ship using UPS, it's only against their policy...
Plan on about $12-15 for 2 bottles or so, depending on the distance and if the recipient has a commercial address. The price for a 6-pack is not much more though.
Comps are great because you can win prizes, but you have to know how to accept the feedback. The quality of the feedback can vary. It's usually best to enter the same beer in more than one comp and close to the same time so you can get a variety of responses and make your adjustments based on a few opinions.