Vertical speed is managed, primarily, with controls.
Lift is proportional to square of speed and to angle of attack¹ and angle of attack can be adjusted using the elevators. Pilot, or automation, use those to control the flight path of the aircraft.
As the aircraft accelerates along the runway, at some point the pilot pulls on the control column, which turns the elevators up and the resulting downforce on the tail lifts the nose off the ground². That increases the angle of attack—because the aircraft is still moving horizontally—, lift exceeds weight and the aircraft accelerates upwards.
As it starts to accelerate upwards, the pilot eases the pull on the control column to prevent the aircraft pitching up further. And as the aircraft accelerates upward, the angle between direction it is flying and pitch—that is the angle of attack³—decreases again, until the forces get in balance. Then the aircraft is in steady climb.
When the aircraft reaches the top of climb, the pilot⁴ pushes on the control column. This causes the aircraft to pitch down, which reduces the angle of attack, and thus lift, and the aircraft starts to accelerate downward, that is slow down the climb. At the point the aircraft is moving horizontally, the control column is pushed back to get the forces back in balance.
Now for the details of operating the controls, the longitudinal stability comes into play. Aircraft are normally⁷ designed to be longitudinally stable. That means the aircraft will pitch up as its angle of attack decreases—which increases it again—and pitch down as its angle of attack increases—which decreases it again. Net result is that when the elevators are left alone, the aircraft will maintain specific angle of attack.
As the aircraft climbs, it will tend to slow down unless engine power is increased, because its potential energy increases and it will be taken from kinetic energy if the engines are not providing enough⁸. And as it will slow down, its lift will decrease, so its angle of attack will increase, but that will cause it to pitch down due to stability. The net result is that it will refuse to climb unless enough power is provided.
Similarly when it descends, it will tend to accelerate, because the potential energy will be converted to kinetic⁹ and that will increase lift, which will decrease the angle of attack and the stability will make the plane pitch up. So it will refuse to descent unless power is reduced.
So in the end, vertical speed is actually managed by thrust levers. Which is also much more easily proven from law of conservation of energy.
¹ Up to critical angle of attack, where stall occurs.
² This is called rotation.
³ Angle of attack is properly defined as angle between the relative wind and the chord line of the wing, but aircraft axis is often used in practice. This makes things simpler as you don't have to consider the angle of incidence—angle between the aircraft axis and wing chord—especially since many aircraft have twisted wings where the angle of incidence changes along the span.
⁴ Or more often the autopilot. While take-off is always flown manually, up above FL290⁵ use of autopilot is required as it gets too tiring and unreliable to maintain the altitude by hand with enough precision to ensure separation⁶.
⁵ Flight levels are defined by pressure corresponding to given altitude, in hundreds of feet, on a standard day. So FL290 is 29,000 ft, but more when it is warm and less when it is cold. The reason is that pressure can be measured easily and quite accurately and flying at sufficiently different pressures ensures the altitudes are also different.
⁶ Originally the minimum separation was 2,000 ft above 29,000 ft due to the lower accuracy of altimeters and lower accuracy of flying as the aircraft move faster in the thinner air up there. But because all the aircraft wouldn't fit up there with those separations, 1,000 ft was allowed provided the aircraft is flying on autopilot and has sufficiently accurate altimeter. This is called reduced vertical separation minima
⁷ Some fighters are intentionally designed as unstable, because it allows faster control response. All such aircraft have computerized controls that compensate this, otherwise it would be very tiring to fly.
⁸ Alternatively, consider that the lift vector is tilted aft, and thus has bigger aft component, which is drag. Physics always has multiple ways to analyse a situation.
⁹ The lift vector is tilted forward in descent, so there is some forward component that accelerates the aircraft.