The only ways to get gems are to:
- Purchase them
- Get them from the tutorial
- Be given them by friends
- Share completed goals on Facebook/Twitter
- Win Colosseum matches
- Win them at the racetrack
- Acquire them from Gemstone dragons
If you just started: At several points, the tutorial will give you gems and tell you to use them. Ignore the prompts, and you'll get to keep the gems.
There is no limit on how many gems you can receive from friends, but every player can only gift 3 gems per day (6 if they have the Dragonsai Gifting Tree). So generally you can get a maximum of 3/6 gems from friends daily - unless you can find someone willing to give you gems for nothing in return.
When you complete in-game objectives called "goals", most message pop-ups will have Facebook/Twitter sharing buttons. Use both to receive 2 extra gems. Some goals themselves also have a small gem reward attached to them.
The Colosseum will win you 0/2/5 gems per day. You can enter one dragon a day, and at most you'll get 5 gems per day back as a reward.
Winning the race gives you a spin of the prize wheel - rarely, you may win some gems. Entering the race is relatively inexpensive and frequently rewards gold, so entering hundreds of times per day is not out of the question, assuming you have enough time to devote.
Every month a rare dragon is available for breeding which matches the birthstone of the month. These dragons can only live on gemstone island. When upgraded to level 10, these dragons produce gems at a rate of 1/week. By default you can have at most 12 dragons on your gemstone island, 3 per island. As Shadur notes in the comments, you can also upgrade each section of the gemstone island for 25, then 50 gems, each upgrading allowing each "quarter" of the island to hold one more dragon. You can get 12 gems a week with a fully stocked, non-upgraded gemstone island, or up to 20 gems a week with a fully upgraded gemstone island. (Note it takes a whooping 300 gems to upgrade all island slots).
Note gems are mostly for extras; speeding up actions, upgrading certain buildings (the nursery) or buying special buildings (the extra breeding cave).
A Bloom Dragon can be bred using Storm + Lichen dragons or any combination involving the elements Plant, Cold and Lightning. Storm + Plant or Lichen + storm also works.
You'll know you have a Bloom Dragon if you get a breeding time of 13 hours. The egg will be rainbow colored.
Bloom dragons started appearing April 2nd and will stop being availible after April 30th. Breed your Bloom Dragons before then or you'll have to wait until next year.
If you breed two of them together you'll be able to breed new Bloom Dragons even after the event is over.
Best Answer
Well, this has been unanswered for a while, but I got bored and I wanted to know the answer to this anyways, so I made a blog post on the subject over at this link. Cheers!