I'd say you want another 1.8 Esteem, especially if this is your first foray into this type of stuff - it should be possible to swap your 1.8 power train into a 1.6 shell, which may or may not be more readily available, but there is more work involved - it can be done, and it's easier if you have both cars side by side, so you can compare and easily swap any parts that may be different.
It shouldn't matter much whether the engine or transmission has failed, or the nature of the failure because you're going to be swapping both the engine & transmission from your car over - another reason to look for another 1.8 is that you can keep what's left as spares.
Now - none of this is likely to be economically feasible unless you're doing the bulk of the work yourself - if you have to pay someone to do it, there will be a significant cost attached, and you might be better off looking for a good used car, and selling the 1.8 to someone who needs an engine for a Sidekick Sport (it's the same engine).
Do you have the mechanical inclination to tackle a project like this, the tools, equipment & shop space that it needs, and since this is your first time out, some one with experience to guide you.
Good luck & go brave - for me, there is nothing as satisfying as wrenching on your own car, and I think you'll find the experience a very rewarding one.