B737 pilot reported autopilot and trim malfunctions during departure climb. Flight crew disconnected autopilot and flew to destination.

Narrative

On Day 0 Aircraft X was operated to ZZZZ airport from ZZZ. The flight crew was me as Capt. & Person A as FO. We took off from ZZZ around XA35z.On departure approaching 4;000 feet; I noticed that the airspeed became erratic approaching 240 knots. While monitoring the instruments; the airspeed stabilized to 240 knots as we continued the climb. About a minute later; the airspeed rapidly approached 270 knots; making us disconnect the autopilot. After the disconnection; the controls were very heavy with an aggressive nose down trim situation. While trying to ascertain if there was a trim runaway pending; FO also grabbed the controls; to try and pitch up the nose attitude of the aircraft. The airspeed of the aircraft was very hard to stabilize until the aircraft passed through 10;000 feet. Air Traffic Control; noticed our climb rate was reduced; as we tried to ascertain what was going on. ATC called us to check if we were okay. I told ZZZ departure; that we had an Autopilot and Trim issue; that we were working on and it looks like we have it under control.I then tried to use the other autopilot; where the aircraft continued to do the same thing; so we clicked it off. FO suggested that it may also be something going on with the Auto-throttles; so we disconnected them as well. We had no caution lights to run the QRH; so we flew the aircraft to ZZZZ airport with it off. The command bars were also giving us erroneous information as well.When we landed; we discussed the situation and flew the aircraft back to ZZZ airport. Where it would be written up and discussed with maintenance as well.I am writing this report because; it appears that from talking with maintenance and my First Officer. This aircraft has been known to have these kinds of issues. Since I'm one of the new guys; I guess that's why I keep getting this aircraft to fly. I suspect the senior pilots know the history and refuses to fly it. I truly believe that this aircraft is unsafe and should not be flown; until maintenance and the company can figure out what's wrong with it.