Changing Theme

if I change the theme, all the posts will be there, is that correct?

That’s correct. The posts, comments, pages are in the database and are not effected by changing themes.

I would like to backup the files just in case, but we have more than 400 posts and I just want to make sure none of the posts, comments, and views are missing.

That’s a really good idea. 😉 Give these a read.

http://codex.wordpress.org/WordPress_Backups
http://codex.wordpress.org/Backing_Up_Your_Database
http://codex.wordpress.org/Restoring_Your_Database_From_Backup

There are plugins that will do backups on a scheduled basis but doing the old fashioned way is the best way to learn in my opinion.

Once you have your backup strategy in place keep them somewhere safe. Routine backups will save you hours of grief someday.