B737 NG Captain reported they received intermittent SPD LIM and AUTO SLAT FAIL warnings at cruise altitude. The warnings became steady and soon after the crew experienced a stick shaker and an over speed indication displayed on the Captain's side. The flight crew decided to divert to the nearest suitable airport.

Narrative

Just before taxiing onto the runway to depart ZZZZ; the AUTO SLAT FAIL light illuminated. We taxied back to an available apron area to contact Dispatch and Maintenance Control to work on the issue. After conferring with Maintenance Control we cycled the circuit breakers that Maintenance Control instructed us to cycle. This did not fix the problem. Upon further research Maintenance Control informed us that the issue could be deferred. A deferral [was] made and a new release was sent to the aircraft.An uneventful takeoff and initial departure was made.Just prior to level off; the SPD LIM indication in the Captain's PFD was noticed to illuminate occasionally but only for a brief period of a couple-of-seconds.Level off at cruise altitude of 34;000 feet was uneventful. All other indications and instruments were working properly; including all primary and secondary airspeed instruments. As a precaution; we reviewed immediate action items; and follow on procedures of 'AIRSPEED UNRELIABLE' checklist. At this time the only non-normal indications were a single channel AUTO SLAT FAIL light when the Master Caution recall was checked; and a very seldom intermittent SPD LIM flag only on the Captain's PFD. The First Officer was already the pilot flying so we continued with these flying positions.Once we were over the X area; the SPD LIM light illuminated more often and for longer periods of time. It was more steady on; than off. Soon thereafter we experienced a very brief (less than two seconds) 'stick shaker'; and very soon thereafter the Captain's PFD intermittently displayed brief over speed indications. We again reviewed the Flight Manual and checklist for 'Unreliable Airspeed.' At this point all of the First Officer's instruments were normal; all airspeed indicators agreed.We again contacted Dispatch and Maintenance Control and discussed the malfunction. No resolution or fix was offered by Maintenance Control so the flight crew conferred and decided to divert. After conferring with Dispatch and Maintenance Control on the best divert options; ZZZ1 chosen. As the flight continued to ZZZ1; all intermittent issues became more frequent.An uneventful visual approach and landing was made on Runway XXL at ZZZ1; followed by taxi to an available gate. After maintenance made an initial assessment of the issues; it was decided by Customer Service; Maintenance; and Ramp to deplane the passengers and accommodate them all on a later flight to ZZZ.The only frustrating issue during the entire event was the fact that multiple ATC agencies (multiple Center controls as well as Approach Control) did not pass on to each other that we had [priority handling] and were an '[priority handling] aircraft!' Center even tried to vector us north of ZZZ1 for 'at least 4 minutes!'I may not have initially checked onto the frequency as a [priority handling] aircraft; and not until I told Center that the vectors were unacceptable and that we were a [priority handling] did we receive priority handling. Center apologized and said they had not been informed. The same thing happened with Approach Control.The only thing I would have changed in this scenario was to ensure I used the label [priority handling] to ensure priority each and every transmission to ATC.