McAir 2012 What do I need to plug it into modem for faster downloads?

That's the standard plug for Ethernet. According to your modem's specs it has one standard Ethernet jack. The modem also has two RJ-11 jacks for VoIP. If you need more Ethernet connections, then you can connect a multi-port switch to the modem. The modem supports 300Mbps which should handle pretty much all cable providers' service speeds.

A faster modem and/or faster Internet service from your provider.

You need the Apple Thunderbolt 2 to Ethernet adapter. For the cable you need a CAT 5 cable with RJ-45 plugs on each end. Most are found in 16 foot lengths.

Yes, any CAT5 will do. 5CATs, on the other hand, is a handful.