Don't all airspeeds change with weight?
Your instinct is correct; almost all of the key airspeeds — those at which something special happens aerodynamically — vary with weight.

There's a distinction, though, between "speeds at which something special happens aerodynamically" and "regulatory limiting speeds that prevent you from doing something bad."

For example, VNE is a regulatory value; it is prohibited to exceed this speed at any weight. This value includes a safety margin and is determined from flight test data. Nothing special happens aerodynamically at the published VNE speed. (Thank goodness!) The special stuff, like aeroelastic flutter or structural failure, happens at some speed well above VNE, regardless of your weight (as long as it's within the allowable CG envelope of course).

Another example, VFE, is similar. Nothing special happens at this speed. It's just a safety value: as long as you're below this speed, you won't tear the flaps off the plane if you extend them, regardless of your weight.

Almost all speeds that aren't "regulatory limits" do indeed change with weight. Some examples:


VA design maneuvering speed
VX best angle of climb speed
VY best rate of climb speed
VR rotation speed
VS0 landing configuration stall speed
VS1 "specific" configuration stall speed


For larger multi-engine airplanes that you'll encounter later in your flying career:


V1 engine failure recognition speed (or takeoff decision speed)
V2 takeoff safety speed
VMC minimum control speed


(Note that a lot more factors besides just weight can go into these latter speeds, such as atmospheric conditions and reduced-thrust takeoffs.)

For small airplanes, it is common to publish these speeds as single values, measured at gross weight. For many of them there are also graphs in the performance section that show you how the actual speed varies with weight.

For larger aircraft, the performance graphs (or calculations) are always used. For example, you won't use a single "stall speed" value from a handy table in the "POH" of a 747. In these types of aircraft, key numbers are calculated as needed on each flight for current weight.