There are 2 things involved when moving sites.
1) Your domain name – You can always transfer names between providers if you need to. I personally prefer using NameCheap.com to register domains so they are independent of the hosting provider. This way you have all your domains in one place and can switch between hosts with a click of a button.

If you are keeping your current domain name with your old hosting company – you will have to point the DNS (Domain Name Server) records at your new hosting company after the transfer takes place.

2) Transferring the site or building a new one
If your current site is not built using the latest technology such as WordPress then you should consider setting up wordpress on your new webhost and manually transferring the content to the new site from the old site. Then you can point the DNS records at your new webhost and your domain name will now link to the new site instead of the old one.

If your current site is built using WordPress and you want to transfer to another webhost then it may be slightly more complicated. There are some wordpress plugins such as “WP Migrate DB” that can help you with the process. The basic Idea behind moving websites is:

a. Copy WordPress files to the new server
b. Copy the Database ( that wordpress relies on) to the new server
c. Make sure your domain name is pointing at the new server after the transfer

Migrating websites is a full topic on its own, I recommend you do some research before proceeding with something like that.