I think we need a stickied FAQ about the handling thing; the question comes up a lot. Anyway, my technique has always been to just put a hand down in their enclosure for 5-10 minutes at a time during the part of the evening when they're active. I don't chase them or try to pick them up or any of that. I just leave my hand sitting there. Yes, this takes patience. They will get curious and come to see what this big thing in their terrarium is. Usually they lick you, and eventually they decide you're ok to climb on. Once they've reached that point, you're pretty much good to go, but you have to be willing to let them come to you.
As far as the eating thing goes, there's probably not much to worry about there - it's unlikely that a healthy juvenile will starve itself. Shipping may be a bit stressful for them, so give them a day to just hang out, and then toss in some crickets of appropriate size - I'd be very surprised if they don't start chasing the crickets right away.
Ensuring a long life really just entails providing a proper environment and proper diet. Use paper towels or tile as substrate, not sand (sand can cause impactions, which can lead to death). Make sure clean water is always available for them - you may not see them drink very frequently, but it's important nonetheless; sometimes they drink a lot of water in one go.
Provide at least 2 hides, one on the warm end and one on the cool end, so they don't have to choose between security and temperature. The hides should be opaque or nearly so, and should have relatively low ceilings - reptiles feel secure when the walls are close around their bodies. Proper temperature gradient - the warm end needs to be between 88 and 92 degrees F, and the cool end should be about room temp. A humid hide is also a good idea - it helps them with shedding.
10 gallons is theoretically sufficient size for one gecko for its entire life, but mine lives in a 20 long (30x12x12 inches) and seems to like having the extra room to roam around - she certainly uses it all.
I'd also suggest picking up a copy of "The Leopard Gecko Manual." It's got a lot of good basic care info in it, along with interesting general info on leos, and can be found for under $10 in lots of places, including online.
I know I haven't provided every piece of info you could possibly need, so please don't hesitate to ask specific questions if you have them. That said, a lot of the info you're looking for can be found in the book I mentioned and/or care sheets provided by many of the breeders out there - theurbangecko.com has a good one, for example, so I'd suggest going to those resources first, and then if you can't find what you're looking for there, by all means, post a question - we're happy to help.
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0.1 Albino Leopard Gecko - Tigger
0.1 Crested Gecko - Pooh-Bear