What type of knife befits peeling/skinning melons and squashes?

You're going to be dealing with large, slippery things, with tough skins.

It's really easy to cut yourself with these.

For softer ones - melons - a serrated knife works well in my experience. A regular bread knife will often be too thin, but a heavier knife with serrations works fine.

For harder items - the fall/winter squashes - I use a shorter-bladed chef's knife or utility knife so that I have more control.

I agree with John Feltz' answer it you need to keep the fruits whole, (as you might, say, with a pineapple) But if you're going to section the fruit, I'd use a stiff scallop-serrated bread knife to cut it safely in the first place (having trimmed a flat side, if necessary, to keep the fruit steady while you do that,) and then something more like a filleting knife to work from the inside, aiming to cut the flesh from the peel, rather than the peel from the flesh.