It's a pretty common type of game - like candy crush for example. You just get adjacent colors next to each other and break them. If there are no remaining moves, the board will be reshuffled (although obstacles such as rocks/ice will remain in their previous positions.)

I'm not sure if there's a more specific strategy post? Each board is random, so it's a lot just up to your intuition to not waste moves. I'll try to explain my strategy... Other people may have different methods/advice though.

The added part is that there are missions - break a certain number of a certain colored block and/or a certain number of ice/rocket/stone. There is a gauge for chest quality which you increase by breaking blocks. The larger the group, the more the gauge rises (ex. breaking 8 blocks at once gives a greater increase than 4 sets of 2)
Additionally, if you complete the board in under a certain number of moves, you get bonus progress on the chest quality gauge. The bonus is generally quite significant but it does depend on the number of moves remaining - ex. 1-3 moves left may not be sufficient for a crystal chest.
When you complete the mission, you get a chest based on your progress in the chest quality gauge and the remaining moves for bonus progress.

For strategy to get higher chests - there are two primary methods.

Strategy 1: complete the mission in a minimal number of moves.
This strategy is the faster way to get through grids, but it is riskier. Essentially, you try to complete the mission in as few moves as possible. Since you don't know what colors are coming next, there is no perfect way to do this, and no guarantee you will be able to complete the mission with enough bonus moves remaining for a gold or crystal chest.

Strategy 2: avoid the mission until you have filled the gauge
This strategy is significantly slower, but very safe. You entirely ignore the bonus moves and just break blocks to fill the gauge, all the while making sure you don't complete the mission, which ends the grid. You can nearly guarantee that you will always get a crystal chest so long as the mission includes a rocket, stone, or ice.
** it may be that you get into a position where your only remaining move completes the mission before you are where you want to be on the gauge, so this isn't entirely risk free, but generally speaking if you are careful you can get a crystal chest each time. If the mission is only colored blocks, it becomes a lot more difficult to go this route, since they tend to build up quickly when you avoid a particular color.

You can of course start with strategy one and go to 2 if you don't get the mission completed in few enough moves, but since you were intending to complete the mission you may be in a worse position to fill the meter without completing the mission.

I don't really have tips for strat 1 since I'm more of a slow and steady type. Just... look at the board and try to see how you might go about getting the mission complete. If there are moves at the top and bottom you want to make, do the top first since it won't impact the bottom. When you make moves lower in the grid, try to be aware of what the impact will be based on the colors you can see above them. This really goes for either strategy.

For strat 2 - rockets are the easiest. If you find yourself with a rocket mission, try and make the biggest possible collection of one color by breaking other colors only, then break them all at once. This will give you a big boost to the gauge to help you get to crystal faster. Just don't break the rocket until you have crystal on the gauge - easy

For ice, you need to not break blocks twice next to the ice. For stone, you need to not let them drop out. I would recommend avoiding the ice all together for as long as possible.

Try to get the ice as low in the grid as possible by breaking blocks under it, but without breaking blocks directly under it, so it is out of your way - it will make it easier to break new blocks as they fall down without breaking the ice. Essentially you're trying to maximize the portion of the board you're actively breaking blocks in. Stone this is also helpful with, but it is also risky because it may make it more likely that you get in a position where your only remaining blocks to break are those under the stone.

Additionally, for blocks of the same color next to the ice, try to prevent more of the same color from touching them. For example, if you have a row of blue under the ice, try and prevent any other blues from touching them. You don't want to break them because they're touching the ice, but the more the blue spreads the harder it will be to get adjacent blocks to break without breaking that group. If half the board is a blue blob you're avoiding breaking, there is less room for blocks to fall with adjacent colors so that you can break them.
However, as you get more progress in the meter, you may do the exact opposite - get a big collection of (keeping the example, blue) that only breaks one ice and get it as big as you can. This will give a large boost in the meter when you break it. Just make sure you can break it without completing the mission in case it isn't enough. For stone, do this as well and then break it with a rocket *under the stone*. Then all you have to do is not break that rocket until you get crystal which is pretty easy unless you are horribly unlucky.

This is going to be pretty long I guess... hope it's helpful. It'll also make more sense once you see the board / missions / moves and play a few rounds. And there is a tutorial for each special block, just read those as they show up.

>>Minotaur: Groom my Wanderers
8 points per action
>>Mermaid: Equip my Wandering Horses for the first time (saddle, bridle and saddle cloth)
17 points per action
>>Gorgon: Age my Wandering Horses with Styx Droplets
1 point per action
>>Faun: Finish training a skill on a Wandering Horse
22.5 points per action
>>Harpy: Congratulate players
5 points per action
>>Centaur:Give my wandering horses mashes
1.95 point per action
>>Chimera: Donate items from the shop to my friends
34 points per action
>>Sphynx: Join missions with my Wandering Horses
2 points per action
>>Cerberus: Donate horses to Ow
10 points per action
>>Scylla: Enter my Wandering Horses into competitions
0.81 points per action

Next event is autumnal mine.

Nalu's price estimate and then her quoted full info is two posts below

Can the divine for the Wandering Horses event be completed without passes??
I think my biggest fear with that event would be the styx droplets bit.

For the Wandering horses event, does anyone know what it means when you have to complete those tasks with Wandering Horses??? Because if so, how do you get started with that?