You should get a swab/urine test for chlamydia & gonorrhoea 2+ weeks after sex and get a blood STI test ideally 3 months after sex (3 months is when it’s most accurate) which usually covers HIV & syphilis. Usually all you do is put a swab up your vagina and rub it around a bit then put it in a plastic tube, for the blood test you’ usually either have your finger pricked or a normal blood test.
Not too sure about US pricing/tests but I recommend checking if your insurance covers it or checking the planned parenthood website x