If your PS2 is still under warranty, and if all you want to do is to play PS2 backups, then swap magic will do the job fine.
If you want to do more (homebrew, PSX games), then you'll probably be better off with a good chip (DMS4, matrix infinity), you can do the same things with swap magic discs, it's just a bit more troublesome.
Judging by the fact that you asked about slide card, I assume you have an old PS2 and not the slim one? It won't damage your PS2 as long as you use it correctly, e.g. don't try to drag out the tray when the disc is still spinning.
If the console's going to be used by children of that age range, then yes I'd advise you to stay away from swap magic, as it's too much of a hassle, and just go with modchip. It will be a bit more expensive for the chip + installation, but it'll be worth the extra money, at least you'll know that when your children play the PS2, they have less things to break :)
My 7 year old uses Swapmagic no problem and has since he was 6.. So I don't see it being a problem.. as long as they are responsible with the discs...
I would suggest going the Swapmagic/Fliptop route. It's alot easier than the slide card and that's what my son and I use....
The only real difference between 3.6 and 3.8 (apart from the price) is that 3.8 has action replay included with it, so you can use cheat codes with your games. But even with 3.6, you can still use cheat codes, it just involves loading action replay as a seperate disc (using swap magic if you downloaded it), then use swap magic again in between action replay and your backup discs.
In short, 3.6 can do everything 3.8 can, some of them may take an extra step or two.
Well, in short, it comes to if you think your children will want to use cheat codes or not, which is probably dependent on what types of games they play? For the sake of simplicity I would get version 3.8, and the only thing they would need to do if they want to use cheat codes is to use swap magic, use the function in swap magic to load the cheat codes, then play their game (the discs itself should have a comprehensive list of codes already - you can always get new codes for new games from the action replay website at www.codejunkies.com).
Not sure how much is 3.8, but I'd imagine it'll be in the range of $5~$10 more than 3.6?
I was needing to ask another question. Is there a place that I could buy the fliplid, swapdisk and some repair parts? I need a new controller port (aka joyport).