What is the best way to farm coins?

You could collect all the 54 golden gnomes and then solve the secret puzzle in the secret gnome room. If you have no idea about what I'm talking, just search the internet for help. Doing this will net you 600k points. You also might consider talking with the clam in the suers and collect the 50 snow globes. Each globe gives you 500 coins and once you got them all, you'll earn another 100k. If you've got all these, go into Solo Play, and play a round of Herbal Assault or Gardens and Graveyards with both teams being composed only out of Sunflowers and Scientist. They will constantly heal and revive their comrades, so you can get a ton of vanquishes, which equal coins and Xp. Xp can be very useful too, since promoting a character gives you 20k coins instantly.

I hope this helps

In the backyard, right in the middle of the map there is a flag you can raise. Its like garden/graveyard ops except it is at your home. Doing this will not only grant you XP but will also give you some money from completing each wave. Remember as you go on, it will get harder but the rewards get better. It is also a great way to complete some daily missions that will often grant you stars. Stars are used to clear rubble where you can put statues at or you can find chest (that usually require 3 - 6 stars  to open). They will grant you money and/or possibly customization's or possibly a big pile of money for you to spend.

Infinity Time usually gives me 25k Coins, and a payout is similar to the Wave you reach or number of shards collected. This method isn't really ideal because you'll eventually run out of stars, but is still viable. Probably works out about 50k per hour, a run to 25k usually takes me just over 30 mins. No exp gained.

Suburbination

Difficulty: CRAAZY

Team Setup: Empty

Quickly capture all the points and just sit there until the game ends. Someone said you can make 100k per hour this way but I timed it and got half of that. This works out to be about 40k per hour if you take into account loading times. Minimal exp gained.

Gardens and Graveyards

Difficulty: CRAAAZY/Hard/Normal/Easy

Team Setup: Sunflowers vs Scientists

Try getting the Sunflowers to group up in the objective by killing a few of them in there, and kill them with Electrican/Electro Brainz, or other classes for slightly less payout. Payout works at around 50k per hour, maybe more. High exp gained.

Not sure how much per hour this gives but it's easy money and it's easy exp, it's a bit repetitive but anyone can do it.

If you prefer Team Vanquish against AI make sure you play Vanquish confirmed, as the orbs give extra exp and well better money as well I think.

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Or play online, any game mode. I wouldn't call this farming for coins as the payout will depend on class compositions and what you do in the game.

Edit: Oops, just realised I repeated @xxSe4l-MLG420xx said too, ah well.

There is a faster way of getting 100k+ coins an hour

The flag of power didn't really give me that much xp

Be sure to finish you Quests every re-set - this will enable the multiplier. 

Run Garden or Grave Yard Ops on setting Hard or Crazy. This has been working for me. PVP doesn't net me as much points for the time played it seems. 

From what I know, it is best to simply enjoy the various activities the game has to offer while racking up coins on the way. I normally do a few Normal/Hard Garden Ops and end up with 200,000 and over coins without noticing. I usually do some Mixed Multiplayer and eventually get the same results without the game feeling like a grind or a chore.

Long story short, this game is not too stingy with coin distribution. Checking the Quest Board and accepting easy quests (usually meeting the criteria that you normally meet anyway such as assist, etc) will grant you easy coins.

Also, there is promoting a character that gives a good 20,000 coins each time they go up 10 levels. Try and play as many characters as possible to get these bonuses plus it even raises your Player Rank!