A Gem Mode game occurs when you play Dash Mode and all the tiles turn green. While in Gem mode you can match tiles any way you want since they are all identical.
Your best bet to start with is when you get the gem tiles from the speed bonus, don't use them instantly. Wait until there's only one more pair in addition to the green gems left and then combine the gems and you only have one pair in your first green gem puzzle. After that you still need to complete two more puzzles when still in the same gem mode.
It's really difficult with mouse, but when you start your second game and get your first speed bonus tiles, if they are Wild tiles, leave them as your last pieces and when the third puzzle opens, combine the Wild tiles to clear the third puzzle instantly.
You need some luck to get the Wild tiles when in Gem Mode. Also if you don't get the green gems as your first bonus tiles, just restart the game since the puzzles will start to get harder in time.