Air carrier Captain reported a stab trim runaway during climb. Crew referenced checklist and returned to departure airport.

Narrative

Shortly after takeoff; maybe 2 to 3;000 feet; we received STAB TRIM and MACH TRIM caution messages. I was pilot monitoring and my FO was approaching his level off at 5;000 feet and needed to be able to trim. I reengaged the stab trim followed by mach trim for him as I knew that was the first step in the QRH and we both monitored it. My intent was to run the full STAB TRIM QRH and MACH TRIM QRH once we finished the initial climb. However; the stab trim aural tone began going off and we were experiencing a stab trim runaway. We complied with the memory items. Since the memory items direct the Captain to fly; I did my best to maintain control over the airplane; however it felt severely out of trim and still very difficult to maintain control. I had my FO begin running the stab trim runaway QRC followed by the QRH and requested priority handling with ATC. I requested to set up for XXR as it was the longest runway; but told them we wanted to remain just north of the airport while we ran our checklists. I was struggling to maintain level flight and the roll control felt like it was hard to control as well. We started getting Trim LWD and Trim RWD caution messages. We decided to begin heading in for XYL instead which was right off our nose and the winds were favoring it. I requested the emergency vehicles be standing by. We couldn't get the ILS signal to come through; but we were visual and I followed the PAPIs. While I did not have time to brief the flight attendant with an NTSBO or notify the passengers; I did have my FO call the FA and tell her we were returning for landing. I used speed brakes to slow down as much as possible and when slow enough; I called for gear down. The QRH had directed us to land Flaps 20 which I called for around 220 KIAS as I couldn't wait long enough to get below 200. Shortly before landing I made a PA to brace. I landed the plane smoothly in the touchdown zone and brought the plane to a stop. I immediately told the passengers to remain seated with their seatbelts fastened and called the flight attendant to see if everyone was okay. She said they were fine so I told her no evacuation was required. The emergency vehicles examined our brake temperatures and we requested for them to follow us to the gate. The remainder of the taxi was uneventful and I gave more information to the passengers when we arrived at the gate. Customer service supervisors took care of the passengers; and I immediately called dispatch and Maintenance Control.Cause: The cause of this event was the stab trim running away. Suggestions: I can't really provide any suggestions for avoiding recurrence of this event. I do think however the training curriculums should include practicing landing the airplane in an out of trim situation.