I grew up in unincorperated Glen Ellyn (next to Wheaton) and had well water growing up. Unless there are pollutants, it's mostly an inconvenience more than it is a health hazard. Jeff provides some great advice below on dealing with the rotten egg smell, and potential for the water to turn your white clothes a "dingy" yellow if it's not sufficiently softened. If the area is unincorperated, you are likely to be on a well - unless city water was offered and the owners paid the additional expense to hook up to it.
The other issue, as stated below, is that wells and septic systems don't last forever - so they are both another potential expense you might be dealing with at some point down the road (and they aren't cheap).
Without knowing the specifics of the property and the associated well, it's hard to answer most of your questions. I would get a thorough inspection of the well and septic obviously, but also get the water tested independently by a lab somewhere. Here's the number to Dupage Enviromental Services - (630) 682-7400 - they can likely point you in the right direction for testing (by someone who isn't trying to "sell" you something).
Hope this helps!
I'm on private well in N. Wheaton and our water is pretty bad although I doubt we would be on the same aquafer. If I had to do it all over again I would have bought a home with city water. It's great not having a water bill but it doesn't make up for the problems.
If you decide to buy a home on well I recommend a few things:
Invest in an iron filter. Most homes will have a water softener but you really need an iron filter as well. Even with the iron filter you'll battle rust stains on your toilet, shower, sink and clothes.
Install a reverse osmosis system for your drinking water / ice maker or sign up for bottled water delivery.
Remove the sacrificial anode from your hot water heater. This will reduce or eliminate the rotten egg (sulfur) smell. Your hot water heater will corrode quicker but it's worth it to get rid of the smell.
I hope this helps.
Prudential Rubloff Properties
145 Danada Square East
Wheaton, IL 60189
(630) 344-5876 (direct)