With most of us in urban areas, I have many customers that have never had well water before, and often have concerns about what they need to know. While not exhaustive, I'll try to post the fundamentals here.
No Matter Where Your Water Comes from
It Does Require some Dough!
The really good news is you won't have a water bill. If you're getting water out of a private well, it's yours, so no fee. Most experts expect the cost(s) of municipal water, or town water (same thing) to go up in the future, so this could be a real positive going forward. The bad news is the water in a well doesn't come out on it's own - it's pumped out, and generally has a small reservoir system to keep the pressure steady. Both of these components can, and do, wear our and need maintenance/replacement over time, so a home owner learns to keep an eye on them just as they must with any other mechanical system.
Many wells do require power to operate, so if you do lose power, once the reservoir is empty, you may lose water too. If you are in an area where power outages are frequent and not short, you may want to consider a battery backup of generator solution. Battery backups can be just for wells, and generators are really a more full service (and expensive) solution. Generally, town water has the services to keep water pressure coming even during outages.
Pressure can be worse or better than municipal flows, but in general is not a concern. As noted above, the systems can help regulate pressure, and there are other tricks (like the faucet heads) that can simulate or increase the pressure should you desire it. All wells should have a pressure test, generally paid for by the buyer, at inspection time to make sure their are no issues here. If there are issues, they can usually be corrected, but depending on the pressure issues involved, they can be expensive, so it good to know the projected cost before you move in. City Water can have low pressure - which requires in house systems to fix - or can have pressure that's too high - which most home owners fix with a pressure regulator.
Well safety is about both biological (bacteria) and chemical factors. All wells should be tested for water quality at inspection time as part of the purchase process. Most issues are easy to fix, but some are not, and generally the tests will give you the peace of mind you seek as to the safety of the water. Well owners should re-test on a regular basis (yearly is often suggested), especially if their communities are in at-risk areas for certain containments. Note that "standard" testing packages oftentimes may not include important, but more expensive tests (like Arsenic and Radon) that I recommend ALWAYS be tested. These relatively uncommon water issues are expensive to fix, and can be health problems to boot. When found as part of the inspection process, it is not uncommon to have the seller mitigate these items. Discuss with your agent which tests you need to perform, and review the quality report diligently. Once you sift through it, you'll be more knowledgeable about your water (a good thing!) and well prepared to move forward with your purchase. Most town water is tested regularly, and is often treated with minerals and occasionally chemicals to keep the water clean and safe. Salts to soften the water and fluoride to help prevent cavities are two common elements that often are added to city water.