Is it a routine test at your family doctor to get tested for acute HIV or do you have to wait a few months after in order to be adequately tested?

Both. It takes time for antibodies to develop, so retesting in 6 wks and 3 to 6 months may be necessary, but for early diagnosis, detection of viral RNA can be implemental in obtaining an early diagnosis, although this is a more expensive test.
Depends on "routine" "Routine" suggests you're asking if standard blood count, blood chemistries, urinalyis, etc will detect HIV. They will not; you need a specific HIV blood test. Some doctors routinely test all patients for HIV, but most do not unless they know the patient is at risk. If exposed or at risk, ask for testing. The Ag/Ab ("4th generation") tests are conclusive any time 6+ weeks after the last exposure.