Chris from Shopify here. In order to get started with a subscription right away, what you'll want to do is go to Settings > Account. Then, there will be a small blue link titled "compare plans" just to the left of your plan details. Clicking the "compare plans" link will bring you to a page where you can select your plan of choice, and your billing cycle. This link has more information if needed: https://help.shopify.com/manual/your-account/manage-account#updating-your-plan
To remove the myshopify.com domain, you will need to mask it with a custom domain of your choice. You will do this by buying a domain through us or a domain provider of your choice, and changing the primary domain to your new custom domain. This link has more information on setting a primary domain: https://help.shopify.com/manual/domains/managing-domains
Hope that helps, feel free to contact firstname.lastname@example.org should you need any further information regarding this. Take care.