I'm going to assume you're using Apache for the purposes of this answer.
First off, is the file being uploaded ok? One possible reason you might have trouble is that the tmp directory isn't writable by the webserver, or readable come to that. Assuming that's ok then move_uploaded_file should work fine.
Create a folder next to your DOCUMENT_ROOT, let's call it "filestore". Make sure it's writable by www-data or whichever user runs apache. Now, you should be able to move the files into that folder. Note they will be owned by www-data:www-data typically - or whatever user and group your server is set up to run as. The reason I put the "filestore" folder next to the DOCUMENT_ROOT folder is that you can be sure the webserver can read the file path up to DOCUMENT_ROOT. Otherwise you run the risk of a folder part way up the path not being readable, and that'll stop you dead. e.g. if you have /usr/local/media as your target folder and /usr/local isn't readable (and executable) by the webserver, you're toast.
If all this works and you absolutely must have you media elsewhere, you can have the "filestore" folder anywhere so long as the whole path to it is read/executable by the webserver. Check each directory in the path.
If these uploaded files are being downloaded by other users via the web then the "filestore" folder only needs to have permissions of 700 since it's always going to be the web server's user which reads them. If other users need access, typically because other software running as a different user needs to use them then you might need permissions to be 750 to allow group members to read (and execute) the directory. You'll also need to add that other user to the www-data group.
For downloads you will need to write a simple script which dumps the file to the browser after doing some authentication checks. That way, you avoid having the media accessible just via http without having any authentication done first - which could make your service into an attractive place for illegal files (copyright violations being the least concern here).