Assuming your ped lets you use sunblock (notice I didn't say sunscreen), it is OK to put it on the backs of a child's hands, just not on the palms and fingers. Always use a "faces" product on children's hands, one that is formulated to not be irritating to the eyes.
I always use no-rub spray sunblocks on my family, b/c they are the easiest to reapply often, and you can see where you have sprayed when it is still wet. My favorite (though I've never used it on a baby, just b/c it wasn't on the market yet) is Ozone Sport, from Hawaiian Tropic, as it does not have a sweet odor to attract bugs or make anyone nauseous. On faces and hands, however, I use a Neutrogena Wax Stick; it stays where you put it, and does not burn the eyes. The most common reason why a child would rub his eyes in this situation is b/c sweat is making sunscreen trickle into them; prevent that and you are ahead of the game. Spray sunblocks contain alcohol, so they can have a drying effect on the skin; be sure to put moisturizer on your children after situations when they have been using spray sunscreen.
My favorite sun-protection gadget for babies is a classic: I always carried a couple of plain (not pre-folded or quilted) cloth diapers. They are thin enough to let air through, very soft and absorbent, and large enough to be used to drape baby from the sun in many situations. They are also useful as giant bibs, cooling blankets when dampened, and to clean up serious messes. They also can be used as a spray-shield to make sure that you don't spray sunblock in a baby's face while doing the rest of his body. Keep a couple in a ziploc in your diaper bag; you'll be amazed how often they come in handy. (Note that a cloth diaper will NOT completely block UV rays; it is possible to sunburn through light fabric like this. However, it will cut down on the glare and heat.)
I second the sunhat recommendation; DS hates hats as a rule, but he doesn't mind wearing them at theme parks.