Air carrier Captain reported erratic airspeed indications and unstable flight during initial climb. Flight crew regained control and continued to destination.

Narrative

Departing ZZZZ there was convective activity around the airport. We originally taxied to runway XXL for departure; while holding short of runway XXL ground said they were not using XXL anymore and offered us runway XY; which I told them we were overweight for; they then offered us runway XZR. We sent for a new release that included XZR and we received one with X; but because the crosswind was close to 25 knots and varying greater I called the dispatcher and requested a release for XZR with Y; which we received. We were cleared to taxi to runway XZR. Ground let us hold short of 1 on taxiway 2 because we told them we had to load our numbers. With the parking brake set we updated the FMS; including building the XZR engine out departure procedure and a return to XZR; briefed the new runway and the ZZZZZ departure; reviewed the release; we discussed the weather and the possibility of a windshear and how we would mitigate that threat; we reviewed the ZZZZZ1 departure procedure off XZR and emphasized to bring the power back to climb power at 1500' and clean up at 3000'; then we ran the runway change checklist; then we continued our taxi to runway XZR. We ran the Before Takeoff Checklist and we were soon cleared for takeoff runway XZR. We were cleared into position runway XZR and then rushed by the tower for an immediate takeoff because an aircraft was on a 6 mile final on the ILS XZR. I looked at the weather radar and didn't see anything significant along our route. We took off and climbed out and the first officer who was flying turned on the autopilot @ 400'. We were initially cleared to climb to FL250.From here on this is the best i can recollect; the rest seemed to happen very quickly.At 1500' CLIMB flashed and the first officer brought the power back to climb; at the same time I was looking at the radar to make sure we were clear of the convective activity and we also received a radio call from ZZZZ Departure. At about the same time @ 1500' I noticed the airspeed was getting slow so I said 'watch your airspeed'. The autopilot was not keeping up and pitching over as we would have expected to regain the airspeed. The first officer pushed the power up to TOGA. The airspeed started to increase and then it was sporadic fluctuating below VFE and right above VLS (Minimum Selectable Speed); the pitch was pretty steady between 5 and 10 degrees; this continued for a few seconds. At one point the airspeed surged rapidly close to VFE and I retracted the flaps from 2 to 1 to avoid over speeding the flaps. The airspeed was sporadic increasing and decreasing rapidly. The flight conditions were pretty stable and the other flight instruments appeared normal (altimeter and VSI were both indicating a steady climb and not sporadic) which led me to believe it wasn't a windshear; the only thing erratic was the airspeed indicator. We could not understand what the airspeed was doing; both of us commented on how erratic our airspeed indicators were; then the first officer said 'help me'. I said I think we have unreliable airspeed and called out the immediate action items that the first officer executed. 'Disconnect autopilot; auto-throttles and flight directors; pitch 10 degrees; climb power'; during this the airspeed accelerated again; and the relief first officer unannounced moved the flap handle from 1 to 0 (which I didn't see because I was confirming the first officer set climb power). I told the first officer to maintain our current heading; I tried to tell ATC this but I wasn't sure ATC understood and I didn't want any traffic conflicts so I [requested priority handling] so we could fly our heading and keep the aircraft stabilized. Soon after while maintaining 10 degrees pitch and climb power the airspeed stabilized and we re-engaged the autopilot; flight directors and auto throttles. After getting above FL180 and away from the weather I expanded my team and called ZZZ Radio and spoke with the dispatcher; maintenance control; the duty pilot; and a specialist to discuss what happened and whether continuing to ZZZ was appropriate which we all decided it was. After the call; Captain A (relief captain) and I agreed that he and his First Officer would be the flying crew and they would complete the landing in ZZZ and my First Officer and I would be the relief crew.Additional Information:After the event the relief crew said they thought we overspeeded flaps 1 by 5-6 knots for < 3 seconds.I'm not sure if it was a contributing factor but later when we discussed what happened as a crew; the relief first officer brought up that the first officer (PF) during the airspeed fluctuation event moved the power from TOGA back to lower thrust or Climb (at least 4 to 5 times). The first officer confirmed that he brought the power back from TOGA to either lower or Climb when the airspeed would increase so as not to overspeed the aircraft. I only noticed the first officer move the thrust levers a couple times; which is when I had him set Climb power and leave the thrust levers there.After the event I noticed in the FMS the predicted landing fuel went from 24;000lbs that was displayed right before we took the runway to 15;000lbs. After discussion over the radio with the specialist we discovered that the 3 step climbs we loaded during pre-flight had dropped out; We reloaded the steps and the landing fuel showed correctly with 23;700lbs. I'm not sure if this was related to the event.