just carry a potion of cure disease. when you're first infected it's the same as any other disease and can be cured. you only turn into a vampire if you don't cure it within 3 days.
Keep an eye on the effects on your character under the magic tab. Pray at the shrines for a cure and added benefits. I use the Shrine of Talos all the time for shorter waits in between shouts. Keep or make Cure Disease potions on you. The shop ones can be given to and used by NPCs who ask for help after being bitten (and you), and the homemade ones can be used by you. You don't really have to avoid any area as long as you can cure yourself. You do have time to do that.
Other than avoiding Vampires, keep an eye on your Effects and look for the disease.
I have to get in the habit of checking my character when I leave a lair or battle with a Nightwalker. I forgot one time and after waiting someplace for 24 hours I was informed that I was hungry. I got the disease a couple of game days earlier. I had to reload.
I became a werewolf to avoid that and all of the other common diseases. Especially since transformations are voluntary and I can ignore it when I want.
Shrines also cure all diseases, so just hit up a shrine whenever you're in town if you're not picky about blessings. There's a Shrine of Arkay in every hold capital except for Morthal and Dawnstar, so there's that.