The steps as outlined by GlowHost are what you have to do but...
It sounds like you are new to web design.
Your current host probably isn't going to give you a lot of help in moving the site. Who knows what it is. If it's straight HTML, no problem. If you have FTP access your problem is solved, just follow the steps from GloHost. If you don't have FTP access you can use WGet
GNU Wget to get it. This excludes any copyright issues.
If it isn't straight HTML you may have problems. Many of these sites use a database to store the information. When a client makes a page request, the server skins the data onto a template and serves up the page. They won't give you the server side programming or the database. You can still use WGet but it may not be perfect. About a year ago one of my customers wanted to host somewhere else (company breakup) I didn't give the new guy my programming or database design. He used WGet and got almost all of it. A half hour of my time and they had a static HTML website.
If you are as new as you sound, I would hire a web designer to help with the site and educate you. There are a lot of issues like Search Engine Optimization besides just having a functioning site.
I hope this helps.
Enjoy Bob