O_O Well first of all, you shouldn't be using your fingernails at all!! I was taught how to use contacts by an actual eye doctor, but I didn't even consider that some people buy contacts for cosplay without that kind of instruction... I wouldn't recommend putting anything in your eyes until you know what you're doing with those things. O_O
At any rate, I wouldn't say you were just clumsy -- figuring out how to get contacts in is hard! It took me quite a while to get the hang of it. o_o
That said, taking them out was always an easier process, I think. It's a bit hard to verbally describe the whole process, but basically what you have to do is use the pads of your fingertips (not the nails!!) and just slide your thumb and index finger from the corners of your eye to the center of it, to sort of pluck the contact off the center of your eye. It's a bit tricky to get used to at first because, well, most people aren't used to touching their eyeballs like that -- I know I was always too gentle while I was still getting used to it which made it hard to do, and I didn't get better until I became more comfortable with doing it and wasn't as hesitant about it.
I think that getting them in is a lot harder because not only is it hard to get used to touching your eyes and not being too hesitant/gentle, but there's also the fact that eyes tend to not enjoy foreign objects being rammed inside them. o_o Like, sure, contacts are designed to be comfortable sitting on your eye once they're in, but, it's the process of getting them in that's hard. Everything that surrounds your eye is there to protect your eye and prevent things from touching it -- which means that you have to basically hold your eye open so that you don't blink or squint or even half-blink. And for me that was always one of the hardest parts -- keeping my eye held open well enough that my eyelids wouldn't try to bat the thing away before I got it on right. This is made even more difficult by the fact that your eyes will probably water at least a little, which means that it'll be harder to keep holding your eye open, because having wet fingers and eyelids makes keeping your grip nigh impossible. >_> I would definitely keep a towel on hand to wipe off your fingers/eyes if you run into that kind of problem.
Oh, and then there's also the problem of making sure the lens itself has enough contact solution on it to be comfortable on your eye, but also not too much solution so that it sticks to your finger... it can be tricky to get that balance down. Which is why, yeah, find someone who can teach you how to contact lens. o_o
The whole blinking/eye-watering thing is sort of present when taking them out, but it isn't nearly as much of an issue, because if you blink mid-way through taking it out, then it's a lot more likely that your eye will push the thing out rather than back in. Getting them in is a lot harder in that regard, because it's a lot easier to blink or half-blink and bat the thing away before it even gets near your eye. :/ There's also the problem of holding your eye open wide enough, which can also be hard to get used to, and also isn't much of a problem for getting them out just because getting a grip on it and slipping it off isn't as difficult as making sure you're giving your eye enough surface area so that the entire lens can make contact with it without letting the edges of it be blocked by your eyelids.
...But yeah. Hope that answers your question. XD
