Is there some way to download everything from FTP?
Yes, but you'd need to (or ask someone to) install an ftp server and configure it for you. I'd suggest against this though. FTP sends passwords as cleartext, and as far as I know, there's no common SSL wrapper (like STARTTLS) for it. FTP therefore is a security hole, unless you plan on wrapping in a secure VPN. I suggest using sftp, which is essentially an ssh connection that responds to ftp like commands.
Is this a dynamic server? Does it have PHP or CGI or any other server side code? If so, HTTP requests can only show HTML as the code is run, not the underlying source. This is not a backup. You should backup the source with sftp.
As for MySQL, again, you're not doing a true backup. You should get familiar with
mysqldump or find someone to run it for you.
As far as what to do with those files, backing up to git will take some scripting on your part. Usually what people do is backup all files (including your db dump), and keep n days worth. If you backup to source control, now you need to back up your repository and worry about corruption. Usually cleaner to just keep files around. Your source code should be in source control anyway, and it should match what's out on the website.
I personally would feel more comfortable with 'If i untar this file it's my website' rather than 'if my git repository is not gorked i can find the tag for this day and it's my website'. In an emergency situation (and all backup restores are) complexity is not what you want.