A PCR will generally have around a 99% sensitivity and a 95% specificity. That depends on things like the nasal swab going all the way to the back in a deeply uncomfortable way and if they only get the front of the nasal cavity (with Covid-19 in particular) it might be just a 95% sensitivity.
Antibody tests will generally have a much lower sensitivity but can detect whether you had the virus in the past.
Dr. Fauci said the only thing you can count on are positive test results, so I would assume false positives are much more rare, but I'm not qualified to say. I do think it's safe to discount a negative test result, especially if you also test negative for flu A/B but still have flu like symptoms and difficulty breathing. Hopefully soon, antibody testing will be available and much more accurate.
Antibody tests will generally have a much lower sensitivity but can detect whether you had the virus in the past.