The server where you bought the domain probably offers hosting services but usually at a cost. If you want to do it for free, have your domain redirected to whatever free web service you decide to use for your website. Remember, free sites are limited in the number of pages and can have a lot of ads. On the other hand, hosting your own website as in Method 2 above, or having your site hosted where you bought and keep the domain, allows you to pretty much create as many pages as you want, without any advertising you don't put up for yourself.
Moving to another website consists of transferring the website’s files and databases, configuring your site with the new host, and directing your domain’s DNS to the new host. Once you pick a new site host, they can usually help you out with this process. The cost will depend on the host you’re switching to, but it will probably be anywhere from $150-$400.