It's not possible to change the "StoreName" of your http://StoreName.myshopify.com address.
Don't worry though, most shops don't even use their myshopify.com address. They make their own custom domain like iloveshopify.com. Makes your shop look more professional and independent :)
No one will see the myshopify.com URL if you get a custom domain. Not even yourself (unless you go digging in your DNS & domain tab).
You can get a domain two different ways:
1. Shopify domains (we take care of the whole set up so it's super easy):
2. Register a domain with any other domain provider.