I was hosting a 40MB file on my VPS and just uploaded it into a standard folder, when people started downloading the file, around 30 at the same time, it started crashing apache and basacilly the whole server.

I was suggested to move it to a public FTP area (eg. : ftp://ftp.sitename.com/filename) so it wouldn't affect the server as bad.

How can this be done? And is there a way to upload files using a standard FTP client?

Yes, you can upload them through an FTP client. It's pretty easy. A good client is at http://www.cuteftp.com - Also, when your asking how you move it.. Do you mean in SSH/Telnet? If so just mv currentdirectory/file.zip newdirectory/file.zip
:)