I'm new to Linode, although I know my way around Linux. I've setup a basic Debian LAMP server and have the first website online.

Right now everything is on the same disk image: server, files and database. I want to use different disk images for files and databases. I have 20GB space, use 2.5GB for the server, 5GB for files and 2.5 for databases. I don't know if it's useful to split files and databases, but this is my idea. This is the current setup in the Block Device Assignment:

/dev/xvda: debian server
/dev/xvdb: swap
/dev/xvdc: files
/dev/xvdd: database

Now I want to mount or access the files and database images, and it's not clear to me how to do this, and how to do this automatically on a (re)boot. I thought this would be done by the manager, but I haven't figured it out.

No, you can't use the web interface to have them mounted automatically.

Read up on /etc/fstab and how to add entries there.