What kind of Ethernet cabling would I need for through the attic - And what cat?
You should only need one cable for the router. Your computer, printer and any other wired devices  can all connect to the router.

As far as cable type, I would install Cat 5e but you could future proof a bit more with Cat 6. Just a question of cable and jack costs - the work is the same.

I normally use plenum rated cable everywhere but I do commercial installs. That may not be necessary for your attic. 

The most important thing is to wire jacks instead of putting connectors on the ends of a long cable. A little extra work but far more professional and reliable. You will need a jack (Cat 5e or 6 to match the cable) and either a wall plate and electrical box or box eliminator or a surface mount box on each end. Get a 110 punch tool and you're all set.