First, while it is true that UV light (short or long wave) can hurt eye tissue the danger comes from staring into a UV light source for a prolonged period of time. An occasional glance won't do significant harm. Also, the level of danger depends on the power of the UV light source; a welding arc can "fry" eyes in a short period of time, while a typical 110V blacklight might take several minutes of staring to do even half the damage of a welding arc - and a small AA battery operated blacklight might take all day to get to the same point as a 110V model. Basically, though, if you keep the kids from staring into a blacklight for more than a few seconds at a time they shouldn't suffer any appreciable harm.
There is a wide variety of blacklights available these days, from small AA battery operated ones to 12 watt 110V plug-in models with both long and short wave elements - and even some with mid-range elements. While there are minerals that flouresce in the long and mid ranges, most react best to short wave, so short wave is the one to go with if you can only afford a single element lamp.
I usually recommend getting a 12 watt lamp to anyone who is serious about mineral colelcting - they can't be beat for the brightness of the flourescence they will stimulate in minerals - but in your case you might consider a lower wattage lamp for getting started; perhaps a 4 or 6 watt one powered by "D" cell batteries. Then if the kids lose interest you won't have spent a lot of money on something that ends up collecting dust in a closet. And if their - or your - interest continues and grows, you can always "upgrade" to a better lamp later on. For the really serious collector, a combination 12 watt 110V plug-in and rechargable battery model with both short and long wave elements is the ultimate field and home study lamp - but be prepared to pay over $200.00 for a good one.
As for places to go to collect, it depends on where you are located. Here in the US the primier flourescent mineral locality is the Franklin area of New Jersey - old zinc mines in deposits which have a wide variety of flourescent minerals, many of which are super bright under a good lamp. Out west, many of the localities where opal or chalcedony are found are worth visiting with a blacklight - as are uranium mines and pegmatite mines where the deposits contain some uranium minerals. Wherever you are, the nearest mineral club should be your best bet for finding out about local sites that contain flourescent minerals. On the Table of Contents page here at Bob's there is a link to a list of American Federation clubs. Have a look at it and see if there is a club near you that you could get involved with. That's your best bet.