what can have a varying velocity and varying speed

Answer 1

Kind of a strange question, since every object can have a varying velocity while having a varying speed.

When you talk about speed, you are referring to a quantity (e.g. John travels with a SPEED of 2 mph).

When you talk about velocity, you also talk about a quantity, but you add a direction to it (e.g. John travels with a VELOCITY of 2mph in a northbound direction).

In other words: When you talk about speed, you only need a quantity. When you talk about velocity you need a quantity and a direction.
So, when the speed of an object changes, its velocity automaticly changes with it.

Remember that when velocity changes, this doesn't mean that your speed needs to change as well. It's also possible that while your speed remains constant, the direction of your velocity changes (and so does the velocity itself).

Answer 2

Anything can. Except for light in a vacuum