You have come to the right person on this one! I attended the University of Florida for both undergrad and grad school, so I spent many years figuring out the best ways to travel around Gainesville without a car. Fortunately, there are plenty of options to choose from when you're going from place to place, and each one will likely work for you depending on where you're headed. Some of these include...
Taking the RTS
The Gainesville Regional Transit System, known as RTS, has convenient stops to many of the apartments in Gainesville. From there, it can take you just about anywhere you want to go, from Butler Plaza to The Oaks Mall. What's great about RTS is they also offer several "Later Gator" routes for when you need a late-night ride home (or out to downtown). I definitely recommend checking out the route schedule on the RTS website to find which options make the most sense for you.
Tip: many people have access to ride the RTS for free, so be sure to check before paying for your route!
There are seven unique neighborhoods in Gainesville, and the preferred method to travel them is often by scooter. These are generally favored because parking and traffic can be more difficult to deal with when you have a car. Smaller and easier to maneuver, a scooter is super convenient to get you from Gainesville apartments to The Flying Biscuit, or wherever it is you're headed. You can even check out the scooter distance of your apartment to UF! The city is steeped in "scooter culture," so you'll find that this is a great option for getting around town.
Tip: since there's so much turnover in Gainesville with students at the University of Florida and Santa Fe, you can often find a great deal on a used scooter at the end of the semester
Walking it out
The best apartments in Gainesville are already conveniently located to the hot spots in the city, so when I lived there, I often found the easiest way to get around was just by walking. You don't have to figure out parking or wait on a scheduled arrival time, plus you get in some bonus exercise while soaking in the sun! I would confirm your route with your preferred maps app the first few times you head out to make sure you're taking the best/fastest path to your destination; that way, you avoid walking around the city aimlessly. But hey, if that's what happens, you never know what great local gem you'll discover.
Tip: invest in a solid pair of walking shoes. They don't need to be the latest athletic sneakers; you'll just want something with some arch support to make this viable!
If a car is necessary, you could always use a rideshare service (Lyft and Uber both have a presence here) or make friends with someone who does have a car. Afterall, carpooling is better for the environment! Good luck with your travels, and welcome to Gainesville.