What is the relation between aircraft descent rate and speed?
If the pilot commands a certain glide path angle, any change in airspeed will also change vertical speed, because both are connected by the glide path angle $\gamma$.



Note that the vertical speed is air speed$\cdot\sin\gamma$. Climbing or descending makes no difference; only the sign of $\gamma$ changes (positive $\gamma$ means the aircraft climbs). Also the type of propulsion will not affect this basic relationship.

Since descending reduces potential energy, some other energy will increase. If the aircraft speeds up, its kinetic energy increases until the increase in drag balances the energy gain. Reducing engine thrust or deploying spoilers will limit the speed increase, but for a steep descent, even with idling engines and full spoiler deflection the flight speed will increase.


  Is it the same for aircraft climb rate?


No. In a climb the potential energy must be increased, and the primary source of energy are the engines. Only now will the type of propulsion affect the result: Both the speeds for the steepest and the fastest climb for propeller aircraft are fairly low, so an increase in vertical speed will normally be accompanied by a reduction in flight speed, freeing up more excess energy for climbing.

Jets also need to fly slow for their steepest climb, but need to pick up speed for their fastest climb speed. This is the more pronounced the lower the bypass ratio of the engines is, so fighter jets need speed to climb quickly. Turbofan-powered airliners fall between propeller aircraft and fighter jets and prefer a moderate speed for their best climb performance.

This is all only true in stationary flight. In a pull-up, any aircraft can gain altitude quickly by trading kinetic for potential energy. But this is a transitionary effect - the high climb speed will only last while the aircraft has excess speed to spend.