Just a few thoughts, possibly not what you are looking for:

Possibly combine some cartoon overlay elements to make the existing transitions a little more whimsical?

I'm fairly new to VMS and I don't have VMS on this PC (just having a quick look at the forum during lunch), but I assume you can do some sort of fly-on or fly-off effect for dragging new clips onto or off the screen.

How about adding a cartoon "pull new screen on / old screen off" effect by having a little cartoon element at the edge of the new clip and moving at the same speed - even a static clip art image moving on a separate layer might do something like this.

Make it look as if a cartoon truck or train is pulling the new clip onto the screen from the side, an aeroplane is pulling it up from the bottom right-hand corner, a monkey is pulling the new scene down from the top, a cartoon tractor (or some roadwork equivalent such as a grader, or something like a forklift) is pushing the old scene off the screen, etc.

e.g. Poor attempt at text visual equivalent:

old scene being overlaid <[truck moving on]<--[new scene pulled on<
or
<old scene being pushed off]<[tractor]< new scene revealed

Multiple tractors could combined with some sort of interlaced bars/blinds fly-off or fly-on effect, if you see what I mean?
------->[tractor]>[old scene>
<old scene]<[tractor]<--------
------->[tractor]>[old scene>
<old scene]<[tractor]<--------

For more complex transitions, perhaps combine other cartoony elements with the transition e.g. spray can without spray, then spray can with spray, overlaid on some sort of iris out effect? Possibly animate an overlaid clipart paintbrush layer to match a transition?

Do a straight cut hidden behind some other overlay action?
Several isolated flowers (or other objects) fly on, followed by a solid wall of flowers that moves on to fill the sceren, then exits the other side to reveal the new clip.

If you're willing to put the time into it, you could create fully animated overlays (I'd recommend Moho by Lost Marble, http://www.lostmarble.com as reasonably easy, inexpensive, and capable for this sort of thing. It also does alpha channels for overlay in VMS). I'm actually coming to video editing from another angle - using VMS to join separate animated scenes from Moho.

The only other child friendly transitions I can think of immediately are scaling and movement (translate) combinations - bounce on or bounce off types, or shrink and grow with movement (old scenes shrinks down, anticipates to the right, flies off left; or new scene zips in small, slows to a stop, expands). Possibly with a still image from the clip rather than the whole clip itself.
More examples of keyframing than a "click on this transition" solution.

Regards, Myles.