B737 First Officer reported Main Stabilizer Trim Fail on climb. Flight crew performed an air turn back and made a precautionary landing at departure airport.

Narrative

On initial climb-out after departing ZZZ the Captain asked me to check my Main Electric Stabilizer Trim to see if it was working; so I checked it and determined that it was not working. He informed me that the Stabilizer Trim had failed on his side also; so I proceeded to the QRH and after reviewing the index I ran the Stabilizer Trim Inoperative procedure. Also; at this point the Controller was directing us to proceed to a fix on the departure SID and I told him that I thought we should [request priority handling] and get a vector towards the airport since I didn't want to get too far away from ZZZ airport with a possible flight control issue. The Captain flew the aircraft to landing and I handled all the administrative tasks. I was quite busy. He asked me to load the ILS approach for XXC which I did and then I sent an ACARS message informing the Company we were returning to the airport for landing due to a flight control issue. I spoke with the Flight Attendants and informed them that we were making a precautionary landing in 10 minutes and to prepare for that and that we expected an uneventful landing. At this point there was some confusion on my part because the Captain asked me to check the box and extend for Runway XXR and I hesitated because I wasn't aware the runway had changed. After I questioned him about it; he cleared the matter up and I loaded and extended off the appropriate navigation fix. At this point; we were on final and I had several more tasks to perform and we both decided the proper order would be based on the Aviate; Navigate; Communicate model. So; I deferred making the PA and ran the before landing flow and on recall the Speed Trim Light annunciated. I quickly ran that checklist for that procedure and then completed the before landing checklist. Next; I made a quick PA to the Passengers and then I got our landing clearance and backed-up the Captain as Monitoring Pilot.The causal factor was all mechanical and we reacted to the given situation. However; I felt as though the recent changes to our procedures added a little bit of uncertainty affecting the ingrained habit patterns that I'd developed over numerous years flying the 737. In my opinion this allowed me to go from normal workload to becoming overloaded quicker than if I was better trained on the new procedures at the flight training center. I spent a lot of my off time reviewing the new procedures so I could perform them as the procedures presented them but felt that formal training for such a big change would've been ideal and more proper.I completed all the tasks and checklists required but did get to a point where I was task saturated running all the checklists; procedures and cabin communications. My opinion is that the recent change to the procedures without formal training allowed me to go from a normal workload to becoming overloaded much faster than I would have. Also; I noticed that I ran an additional checklist that wasn't necessary; I ran the Speed Trim Fail checklist that annunciated as I did a Recall as we turned final. The Stabilizer Trim Inoperative checklist Additional Information section states: that after the Stab Trim cutout switches are placed in CUTOUT; the SPEED TRIM FAIL light may illuminate. I believe I was [becoming overloaded] at the point I saw the annunciation of the SPEED TRIM FAIL light so I ran that procedure unnecessarily adding to my workload.