All I have is the right climate.
Here, the more Echeverias, the better. I remove the leaves from the flowering stems (they flower here) and lay them on a dish of cactus soil mix--the plain old packaged kind, not the special hand made mixes that real plant lovers make and experiment with and perfect for their own conditions over the course of decades.
In a few weeks, new baby Echeverias. They're like people--adorable as babies. Unlike most people, they're adorable when they grow up, too. Keep in mind that baby Echeverias need more water and considerable shade--they are not as tough as well-established plants.
Besides propagation from leaves, if the main stem of the Echeveria becomes long and bare, you can cut the rosette off, let the wound dry for a week or so, and reroot the rosette. An expert recommends suspending the rosette just barely out of reach of a layer of growing mix. The rim of the pot can act as support. The rosette will send roots reaching down towards the mix. When you see a good amount of roots, it's ready to plant in the soil.
If your humidity is not too low, just leaving a cutting or broken stem in full, bright shade can prompt roots to sprout:
Fallen leaves will also form roots, in a protected shady spot:
When beheading an Echeveria for re-rooting of the rosette, if the old bare stem is in good health, you need not discard it. It is possible it may sprout brand new baby rosettes!
These too can be cut off and rooted. Echeverias--the more, the better! Why? So you can make tapestries out of them. Here's a page with Echeveria tapestries. I don't have any of my own yet, but now you understand why I'm propagating so many. Here's a post on a spectacular Succulent Tapestry.
Some other good design tips for succulents here. Another post on Echeveria propagation here.
Update July 31, 2014:
One of my Graptoveria 'Fred Ives' has formed a crest. A crest is an abnormal but attractive growth habit. In 'Fred', it creates a flattened rosette. It's an interesting and fun variation of the "normal" form:
I cut the stem and placed it in a pot of dry gritty mix, and the pot went into bright shade with no direct sunlight. Five days later, roots! Note the circled area in the photo. That bit of emerging root sprouted from where a leaf had been. I pulled some of the leaves off to expose more stem.
Now that it has roots, I'll water it just once, and wait a bit--a couple of weeks, perhaps, to give it time to grow more roots. Then I'll start watering it regularly, once every one or two weeks.
For easy growing, I recommend starting with Echeveria 'Imbricata', which is thought to be a natural hybrid, and Graptoveria 'Fred Ives', a hybrid of a Graptopetalum and an Echeveria. Both are vigorous and not as fussy as some of the species Echeverias.