If you have an insurance agent you trust (Allstate, State Farm, Farmers, etc), call them! It is not as difficult as you think to get life insurance. We have ours through Allstate. Budget or no, term is the way to go. The idea behind life insurance is that if something should happen to one of you, the other would be able to live without your work (whether that is raising the kids or working outside the home, or both). So you need to figure out how much money the other person and kid(s) could live off of, raising the child to adulthood. The insurance company will give you some guidelines around this, as well (for instance, if you make $30K/year, they will not give you a million dollar insurance policy - they don't want you to be worth more dead than alive. :) Ask for a policy that is renewable and convertible (you can renew it at the end of the term, and/or convert it to whole life at that time, if you choose).
Anyway, some insurance agents will try to get you to buy a whole life policy. Personally, I wouldn't do it. They are taking the extra money and investing it on your behalf, and charging you outrageous rates to do so. If you have extra money to burn, put it in an IRA. You don't need your whole life policy to invest the money for you for a worse return than you could get yourself. Also, don't buy insurance on your child, cheap though it may be. Generally, you only want to buy insurance on assets, not liabilities, and as much as we love them, children are liabilities. :)
You will have to go through some medical questioning. In our case, they sent a nurse to our home to take blood pressure, height/weight, she took some blood samples, which were tested for cholesterol (and I'm sure checked for HIV and other illnesses like that), and we answered some basic health/lifestyle questions (do you smoke, do you exercise, do you ride a motorcycle, etc). They did not ask for any doctor's records. However, if you lie and they find out about it, they can cancel your policy, so be careful about that. We did not have any hair samples taken that I recall.