How can I safely connect cable in a service panel?
Stay away from the metal parts above the main breaker. They are hot always. 

If you haven't turned off the main breaker, also stay away from the bolts and buses right below it, and the buses down the middle that the breakers clip into will also be hot.

It helps to use gloves if you can get PPE in this day and age. 

Note the way the last guy runted off all the wires so they can barely reach where they're going right now. Don't do that. Make each of your hot and neutral wires long enough to reach any breaker location in the panel, so you have freedom to move things around later, which you may well because your panel is full. Neutrals also need to reach the breaker in case you add AFCI/GFCI/??CI later. To make it "neat", run your wire along the edge, overshoot past your breaker then double back and pull in the way they have.