B737-400 First Officer reported failure of both autopilots shortly after takeoff that necessitated a return to the departure airport for maintenance.

Narrative

Captain was the Pilot flying (PF). Shortly after take-off; went to engage autopilot; engaged and then disengaged; and would not reengage. We then chose to attempt autopilot B. Autopilot B engaged but autopilot trim was not working which led to the illumination of the STAB OUT OF TRIM light. The Captain advised to pull out the QRH for STAB OUT OF TRIM light. We completed the QRH. The STAB OUT OF TRIM light remained illuminated throughout the entire flight. We then contacted the company where they advised us to return to ZZZ. We communicated with ZZZ ATC that we had a maintenance issue and we're advised by the company to return to ZZZ. During the descent going below 10000 ft. (around 9850 ft.) we may have been slightly faster than 250 kts. as well as slightly off assigned heading. Advised the Flight Attendants (FAs) and passengers of our return. We landed in ZZZ with no further issues. Taxied safely where maintenance addressed the issues. We reached our cruise altitude of 26000 ft. when ops gave us a response to head back to ZZZ. Our landing weight was below max landing weight. Task saturation on both myself and the Captain lead to the slight excess in speed and slightly off on heading. As soon as I saw the issue; I directed the Captain to level off and turn back to the correct heading. They immediately leveled off speed dropped to 250 kts. or less and we continued the descent on the assigned heading.