There are four main roads that take you everywhere you need to go in Gainesville. Archer Rd., University, 34th, and 13th. During the middle of the day, none of these are difficult to navigate. But at peak times early in the morning and from about 4 p.m. - 6:30 or so (maybe longer on Fridays), these can all become unbearable. But it also depends on which direction you're heading. The flow of commuter traffic is pretty obvious, and is easily avoided once you learn the patterns.
My tip for navigating Gainesville would be to familiarize yourself with the back roads as soon as possible. Gainesville is on a grid, so it's really easy to take shortcuts pretty much everywhere you need to go, except for the Oaks Mall. There is really no convenient way of getting there.
Stay away from the 20th/62nd area off of 34th. There are tons of apartments in this area, but the traffic is absolutely horrible in the evenings. Plus they're working on construction right now, so it's even worse than normal. But since you're looking for more family-oriented areas, you probably weren't looking in that area anyway.
I did a couple years of living in apartments in the student areas, but for the past two years my boyfriend and I have been renting a house in a lovely, quiet neighborhood that's still really close to everything we need. There are tons of families in the area, and although I do see other college students jogging around, we never have noise issues and parties are extremely few and far between. I can't even remember the last time there was one in the neighborhood, aside from the celebrations post the numerous National Championships
There are always houses for sale/rent in this neighborhood, so if you'd like to check it out as an alternative to the more pricey Haile Plantation (for the record, Urban Meyer lives in Haile, as well as the President of the University), it's off of Newberry Rd across from the Regal Royal Park movie theater. The neighborhood is well kept, and has a nature trail and a park within walking distance with a jungle gym, swings and various walking paths. Plus, it looks as if they've set up a community garden! It's a couple miles from campus, but is definitely an easy bike ride away. A 15 minute drive to campus (max with traffic); 15 minute drive to any of the other businesses you might want to go to, unless it's rush hour. If you're interested in looking more into the area, I can give you a more specific address to look up. I'm renting through Gore-Rabell properties. You might want to look at their Web site as well!