Jay, Deb and Len have all provided you with good information. I'll add more details so that you understand what you should test for. Remember, the testing is an opportunity for you to confirm that both the well and septic system will provide you with years of trouble free service and also provide you with not only potable (drinkable) water, but with water quality that YOU want to live with.
On the well, you want to evaluate three issues. I'll speak from the prospective of modern drilled wells as that's what's most common in my area and in the new construction that I deal with:
1. Yield: Should be tested to confirm what its yield (GPM - Gallons Per Minute) is and the records on file should give you an idea of the well depth and static level (natural level below surface of ground that water settles at). These two pieces of info give you an idea of how much water your well will deliver. The deeper the well, the more reserve water you might have available for use. In a well with 6" casing, there is 1.5 gallons per foot of water column (drill depth - static level - pump distance off bottom of well = water column x 1.5 = total gallons in ground).
2. Pressure/Output Capacity: This is related to yield because a low yield would limit the best delivery system's capacity to provide you with water on demand. The deeper the well, the larger the pump must be to push the water out of the well when you put stress on it during peak use and when the static level is drawn down. Larger pressure tanks inside the home are generally better in that they will put less cycles on the well pump. If you are planning to add an irrigation system, these two issues need to be evaluated by someone knowledgeable so that you can confirm that the existing well has the capacity to support the load that an irrigation will put on the well.
3. Water Quality: There is a difference between water that is potable (safe for human consumption) and water that is high quality. Learn more about any treatment system that is already installed. If there is an existing water treatment system, its backwash drain line should NOT be tied into the septic system drain. The current public health code prohibits this. The biggest concern I have with water quality is that if the natural water is aggressive, meaning low in PH and other factors are present, that condition needs to be corrected via a treatment system in order to avoid the copper plumbing in the home from getting damaged and failing. If you see green residue in the toilet tank, that's possibly a sign that the copper plumbing may be negatively affected by the water quality. I'll add one test to Len's list and that you might want to test for fluoride if you have young children knowing what the natural level of fluoride in the water is will allow you to set a proper supplemental level of fluoride to provide your children with should you want to do that (it helps with teeth development).
Best of luck with your new home search!!
Greg Hanner, Broker, REALTOR, e-PRO