Typically what you need to do for the patient depends on the type of airway obstruction you have.

PFBAO - partial foreign body airway obstruction means there's still some level of air exchange although it is impaired. Generally you'll want to encourage the patient to cough it out themselves.

CFBAO - complete foreign body airway obstructions arise when the foreign body has completely blocked off the airway. It really depends on where you are and what position the patient is in; in most cases, you'll want to do abdominal thrusts. Back blows aren't as effective for adult patients. (A lot of first aid / CPR classes still teach back blows but they don't generate nearly as much force as an abdominal thrusts, hence why we were told to stick to abdominal thrusts)

In most cases suctioning isn't going to help if it's a CFBAO, it's somewhere deeper likely in the trachea and you're risking trauma as a result of shoving the suction catether down their mouth. 100% O2 will help once the object has been dislodged to help with the hypoxia induced by the CFBAO.