I didn't see a guide for this in the forums, so I figured why not make one myself?
Note: I don't have images embedded because I didn't want to include them, but I can and will add them if people want/ask me to. Yes there are paragraph spaces.

What is Jinxie's Jamboree?
A slot game that's fairly cheap and incredibly easy to play. If Sylestia is working and you have gold on hand, then you can play Jinxie's Jamboree

How do I play?
Have some gold on hand (I'd say 1000 at the minimum) and click the green "Jamboree with Jinxie" button on the bottom. The button will be green/white if you can play. Each turn is either 5 or 10 gold per play. The gold won when you pay 10 gold per turn is double what you win when you play 5 gold per turn (except the jackpot). My memory isn't the best but I believe the 10 gold tier is unlocked once you reach Astryl's Vale. To select a tier, click on the tier you want or see if the tier is green/yellow (rather than yellow/pink), which means it's selected.

How do I win?
When you have either two or three squares with Jinxie's face (the blonde girl) on them in the middle row, you win a ~bonus round~ (it says bonus game but it's not actually a game).

When does the game end?
The game ends when run out of gold or you leave the page (technically). Any lanterns (which are explained later) you have lit will be reset if you get three squares with the guy (I have no idea who he is) on them. Leaving the page to do something like hatch an egg from your hatchery resets the game to its original position of the top and bottom rows (which you don't win anything from) all being Jinxie and the middle row being all diamonds. However, if you had won two or three Jinxies in the turn before you left the page, the button is the "Play Bonus Game" one not "Jamboree with Jinxie" so you can collect the gold from this bonus game, and any lanterns you had lit up will still be lit.
I do not currently know the limit on how long you can be away from the game and still go back to the bonus game, but it is at least six minutes if you use the same window.

What can I win?
The most common prize you can win is gold. The chart at the top of the game is fairly straightforward in my opinion, so I won't really go over what it says. However, I feel it is important to mention that for the pink diamond and trophy squares, you will only win what it says (ex. 1000 or 2000 gold for two pink diamonds, 80 or 160 gold for two silver trophies) if they are lined up on the left.
In other words, if you have two pink diamonds and they are in the left and middle squares in the middle row, you win 1000 or 2000 gold, but if they are in the middle and right squares you don't win the 1000/2000 gold. If the diamonds are, say, in the left and right squares, you win the 20/40 gold for the left diamond but no gold for the one on the right.
If you spin one or two diamonds that are on the left and a trophy, you will win gold only from the diamonds. If you spin two Jinxies and a trophy, you will win the gold from the trophy and whatever you win from the bonus game (explained later).
The position of the trophies, Jinxie's face, and the lanterns don't really matter, as you will always win gold for having a trophy, you need two or three Jinxies to unlock the bonus game.
The lanterns are explained later, as they have their own prize pool.

What happens when I get two or three Jinxies?
When you have two (or three) Jinxies in the middle row, the "Jamboree with Jinxie" button changes to either blue or purple and says "Play Bonus Game". Once you click this, a rectangle pops up and tells you to "Click an Image to Uncover a Bonus Reward!", and underneath the helpful instructions this sentence is a five column, four row chart of Jinxie's face. You can click one of the Jinxies, and will win anywhere from 5 to 100 gold (in the 5 gold per turn game) or 10 to 200 gold (in the 10 gold per turn game). Click "Claim" underneath the Jinxies to get this gold and continue playing.
In the purple version (when all three squares are Jinxie's face), the prizes are all the same except sometimes you will find a treasure chest. It is epic (purple) quality but level 0, so the gold won is likely more than any of the other squares but not over 1000 gold.

What's up with the lanterns?
The lanterns are what you want to spin. When one shows up in the middle row, the lantern at the top of the column the lantern square is in changes from greyscale to its actual colors (what's shown on the square) and the background of the lantern above the game changes to yellow.
You do not win anything from getting a lantern. You win when you get a lantern in each row without all three lanterns at the top are lit up/their backgrounds are yellow. When this happens, the "Jamboree with Jinxie" button changes to an orange/yellow "Play Bonus Game" button.
This bonus game is similar to the one you play from getting two or three Jinxies, but is much larger and has a significantly better prize pool. The chart is much bigger (eight columns and six rows), and some of the squares hide the guy who causes the lanterns to be reset in the regular game. This bonus game ends when you find the guy three times

What can I win from the lanterns? (aka the reason why I wrote this guide)
- Gold (anywhere from 10 to 500 in one square)
- Lesser Philters (500 scales in the scale shop each)
- Lesser Pet Dyes (1000 scales in the scale shop each)
- Regular Pet Essences (250 diamonds each, but are sometimes found when battling and nurturing)
- Lesser Genetic Mutators (1500 scales in the scale shop each)

Is the payout worth it?
I'd say yes. I'm an unlucky person, but rarely (if ever) leave with less gold than I started with, though it's not usually an insane amount more than I started with (maybe 3000 gold if I don't do too well, but I also have about 6.250.000 gold right now), Lesser Pet Dyes and Lesser Genetic Mutators sell for a nice amount of gold each. If you take a chance, you may even get lucky by using a Lesser Genetic Mutator like I was here and get a visible trait

Tips and Other Info
- When you unlock the 10 gold per turn tier, use that tier only. You will win more gold this way
- If you play for too long in one sitting (xxx minutes) without switching pages in Sylestia (no you don't need to leave the page you're playing on), a (Sylestia-made) popup saying "Please Confirm You're Alive!" shows up. Click the named shape and you can continue playing
- Order of images before and after the lanterns
- left: guy, Jinxie, lantern, silver trophy, guy
- middle: silver trophy, guy, lantern, Jinxie, bronze trophy
- right: bronze trophy, Jinxie, lantern, silver trophy (sometimes), guy
- If you manage to get two Jinxies and the final lantern you need to unlock the bonus game, you will play the bonus game from getting two Jinxies first and the bonus game from the lanterns second
- You can refresh the page if you get two squares of the guy to avoid having your lanterns reset, but this does not guarantee you will not spin this again
- You can be on the 5 gold per turn tier, get a lantern, and switch to the 10 gold per turn tier, but I would advise not doing so (idk if this is cheating since you spend less per turn in hopes of getting the lanterns)

Stuff I Still Need Info On
- what people win from the chests from the bonus game from getting three Jinxies (just gold? how much? anything other than gold?)
- which Essences people win (ex. can you win Regular Ferrikki Essences?)
- which Lesser Philters people win (I've won like three in total and two were the same, so idk if you can win all of them)
- how much gold people win from one square from the bonus game after lighting up all three lanterns
- anything I'm missing

also: feel free to tell me to add anything I'm missing from the guide (especially in the other info section)

What can I win from the level 0 epic chests from the lantern bonus game?
- Gold

What People Have Won
981 gold
1280 gold
981 gold (again)

I read most of your post and I was looking for something specific. How do unlock the bigger/better/more expensive tiers? I've been trying to figure this out and the other day I finally unlocked tier 2 but I have no idea how or why.

To be entirely honest, I have no idea how the tiers are unlocked. I've been playing Sylestia for 35 days so I don't really have much experience with the game. As far as I know, only the 5 gold per turn and 10 gold per turn tiers are unlocked, as those are the only two I have unlocked despite being at stage 6 for the lost grove (the furthest anyone can be as of now).
I haven't found any guides on Jinxie's Jamboree so I don't have confirmation for this, but I believe that the 10 gold per turn tier is unlocked once you reach Astryl's Vale/Ethernia

Alrighty thanks. :)

I just got a Purple Brittle Relic of the Assassin from one of those chests.

I realize this is a late reply. Short answer- the higher tiers are permanantly locked.

Long answer: the game is currently weighted in the player's favor, but not profitable enough for it to be superior to most forms of grinding. If the payout were multiplied 25x, that would change. As such, Krin has said he has no intention of unlocking them at any point in the future. I would assume this is set in stone, barring rampant inflation. (Actually, Krin said that if he did he'd have to modify the game so that it's house always wins not house alway loses, but then there wouldn't be a point in playing it so he's not going to go to that trouble.)

Hm interesting. Thnx!

for the lanterns game you can get 3 heads in a row and therefore get NO gold. (It happened to me twice)