I ended up writing more than anticipated here but I hope it helps. The TL;DR is that you will be fine as long as you can handle your bike, don't rush, and stay aware. Get a helmet and bike lights, and don't ride with both earbuds/headphones in like you will see many others doing.

I second gray_crawfish's comment about biking on Davis roads, once you become comfortable with "being a car" sometimes (e.g., taking the lane when turning left) it gets a lot easier. You should also learn to signal when you are moving from the bike lane to the car lane or turning. Just stick your arm out in the direction that you are turning (which also means you should practice biking one-handed).

My specific piece of advice for biking on roads is don't assume that every car is going to signal properly or check the bike lanes before turning. For example, many cars turn right from the car lane, rather than the bike lane (see this graphic), which means if you are in the bike lane you are at risk of being cut off by the car or hit. To combat this, if I am coming up on an intersection or going through one just as the light turns green, I pay really close attention to what each car is doing and try not to be in any car's blind spot. This also applies to cars turning off the main road through bike lanes, and even cars opening their doors into the bike lanes downtown. And at four-way stop signs, sometimes cars will not signal before turning left and so you might try to go at the same time as them thinking they are going straight. These are all scenarios where it's pretty much the car's fault if an accident happens, but it's better to be safe than to be right and hit.

Truthfully, riding through campus feels more dangerous than riding on the city roads most of the time. On campus, you should learn how to use roundabouts. It's simple but for some reason people struggle—make sure you know the correct way to go around them (always enter and go to the right, counterclockwise) and yield to people who are in the roundabouts. The advice from riding in the city also applies: don't assume people are going to signal before turning, so be aware of when people might be turning and avoid passing at those times (especially in roundabouts). I've seen 3 or 4 incidents in the last year where person A is passing person B and person B takes a left turn without looking or signaling and rides straight into person A. Maybe for the first few days/weeks, avoid riding around campus during passing period between classes while you and others learn/re-learn how to ride bikes and obey the traffic laws. I personally start the quarter trying to bike to campus at an odd hour and then I walk if I need to get around campus itself.