Unfortunately, there is a charge of $18.00 to transfer your character to another server, and not all servers are open (particularly the newest servers where the economy is not as developed). You are also limited in the amount of gil you can transfer as well, so if you have a stockpile, try to dump them in to tradeable goods like materia or mats.

As for servers, you might consider Shiva, Lich, or Phoenix as those are the most popular EU servers. I don't have any actual experience on them so you may want to create a new character there first to test the waters and community.