Is the internet just the UCSD network, or does each unit get their own internet router?

It's a joint account with Spectrum, you get equipment (Router, cable box) assignment to your unit!

The internet is not part of the UCSD network. The internet used is spectrum. If you're moving in to a new apartment then you'll have to set it up with spectrum. But if you live with roommates who have the internet set up already, then you are good to go. It is advised for grad students to switch the name of the holder for the internet service since grad students only have a 2 year contract. The internet bill is included in the rent.