As with many aspects of COVID-19, it's hard to give an exact answer of when you should get a COVID-19 test, however there are pretty good recommendations.
The CDC has stated that those who have symptoms of COVID-19, have had close contact (within six feet of an infected person for at least 15 minutes) with someone with confirmed COVID-19, or have been asked or referred to get testing by their healthcare provider, or local or state health department should get tested. Symptoms include fever or chills, cough, shortness of breath or difficulty breathing, fatigue, muscle or body aches, headache, new loss of taste or smell, sore throat, congestion or runny nose, nausea or vomiting, and diarrhea. Most health officials and doctors will agree that if you feel you have a reason to be tested, then you should be tested.
If you find you are in a bit of a grey area, for instance you came into contact with someone who is showing symptoms of the virus but has not been tested yet, you should instead call your doctor about next steps to advise you on the best plan of action. However, if you are concerned you may have been infected, there is no reason not to get tested.
On the other hand, you do not need to repeat testing to prove you have been “cured” of the virus after infection, because it is not really reliable. People can be positive on a test after they are no longer contagious. Testing like this, or when you do not need it for one of the previous reasons mentioned could put a strain on the system and may take away the ability to test those who truly need it.
A more realistic possibility is to get tested once or twice after exposure. Often, a person with the virus would test positive around three to five days after contracting it. The CDC says the virus has a median incubation time of four to five days. Which is about the average amount of time it takes for symptoms to develop, however the incubation period could be anywhere from two to 14 days. If you get tested the day after potential exposure there is a high chance of a negative test even if you were infected. If you get tested twice after exposure try to space the testing out appropriately and wait at least five days after exposure before getting tested the first time.
If you do get tested you should quarantine at home pending test results and follow the advice of your healthcare provider.
The CDC does not have recommendations on who should get an antibody test, and instead refer you to state and local guidance.