- Drink a Potion of Cure Disease; or - (free option) activate any shrine of the Divines (like the open-air one in Whiterun, with the screaming guy in front of it) and you will see "all diseases cured" on the upper left, removing the illness from your active effects.
Always helps to have a few Cure Disease potions on you, especially if your dungeon diving.
I have gotten more diseases from Conjured Familiars than from wildlife, but any animal can give you a disease. Vigilants of Stendar also carry Cure Disease potions that they will give to your character if you are diseased when you run into them.