Having PCOS should not affect the accuracy of a home pregnancy test. Pregnancy tests detect human chorionic gonadotropin (HCG), which is produced when there is a fertilized egg in the uterus. You don't have it in your body unless you're pregnant. That isn't to say that the tests themselves can't provide a false negative or false positive, but having PCOS won't cause the test to be wrong.
Some women with PCOS do get pregnant unexpectedly--your doctor should not have told you that you didn't need contraception when you didn't want to become pregnant! We don't necessarily ovulate regularly, but some of us do ovulate occasionally. Anytime you ovulate and don't use contraception, there is a chance of pregnancy. We just have no way of predicting if/when we'll ovulate. In the future when you're no longer trying to get pregnant, consider a non-hormonal method of birth control--I've used a diaphragm and would recommend it.
As far as trying to get pregnant, the first step for many women with PCOS who have trouble conceiving is metformin (Glucophage). PCOS is often caused by insulin resistance, which causes insulin to accumulate in the body and interfere with ovarian function. Metformin is a diabetes drug that helps with insulin resistance and can return ovarian function to normal. It doesn't work for everyone, but some women can ovulate by taking metformin.
If metformin alone doesn't work, it's often combined with fertility treatments like Clomid. This is a true fertility medication (unlike metformin, which is not a fertility drug). Every patient responds differently to treatment--for some women met alone works, some respond to Clomid, some have to pursue further treatment and some need no help at all.