How/where can I farm gems?

The best way to minimize the time needed to get higher quality gems (which is ultimately what you will go for - even if you want to farm low-level gems) is to go to higher difficulties as fast as possible, since stronger enemies will have a higher chance of dropping higher-level gems.

This does, of course, mean that special opponents such as champions, bosses, or the treasure goblin have a higher chance of dropping them, but unfortunately, I don't think there is a place with tons of champions always walking around. Basically, you're up to your fortune here in how lucky you are to find gems.

The thing with gems is that there are a lot of different quality levels with them this time around. As far as I'm aware, there are 17 of them. If you spend a noticeable amount of time in one area just to grind for gems, you're most likely just wasting you're time, because the same time spent on advancing in the game will get you more higher quality gems earlier on. e.g. at the beginning of nightmare difficulty, monsters still drop mostly flawless or normal gems. At the beginning of Act II already, they're dropping mostly perfect gems and if you're lucky, radiant gems. This means that farming for gems is mostly a waste of time.

In my opinion, the best place to farm gems quickly is act 1 inferno. In act 1 inferno, flawless squared gems are dropped, and those are the highest gems that can be dropped.

If you'd go to act 4 inferno, the same quality of gems would drop, but the pace at which you could kill enemies would very likely be much slower. Stay in act 1 because monsters are relatively easy there.

Obviously, keep your eyes open for resplendent chests and treasure goblins, but that goes without saying.

An update, as much of this info is currently outdated.

It seems that running the bounties that kill bosses (any diablo-head icon next to the waypoint on maps with the associated bounty, such as Belial, Magda, etc) is the quickest and most profitable way to farm gems solo.

I have finished my third boss and have 20 marquise and 3 imperial. Note: this is solo with 0% MF at t16. You're always going to get better drops at more difficult levels.

I find that just encountering elites and whatnot do not guarantee gems, and I prefer not to bother with [g]rifts for normal gems.