Yes, you will need to attempt to get the correct visa.
Generally student visas will grant residence to accompanying
family but no right to work (ie F1 and M1). My understanding
is that the J1 will permit a spouse to apply for right to work however
the J1 is likely to be more difficult to obtain - its an exchange program.
Spouse and children will generally be recognised, I doubt that
partnerships or partners would be recognised as applicants.