SO, using the casino is the easiest way to build the capital to buy out the showroom, but you can also do it if you have not picked up the casino as a possibility.

so to buy out the initial items in the storeroom, you're looking at between 130 and 220 coins. tbh it's probably usually around 160.

the trophy itself costs another 100 coins and you do have to buy it the same run to get it

so you need 230 to 320 coins, realistically.

Upping your allowance is the first thing you can do, and it's nice in general to be able to afford things off the bat. You get the allowance by going behind the statue in the cloister and picking up the first allowance token.

In addition to allowance tokens being in the puzzle boxes and areas you need to work to get to, I can think of three optional ways to do so

The first two you can do regardless of your life choices, the third requires picking a the Quest Bedroom upgrade to the Guest Bedroom, which ups your allowance 2 coins every time you make it into the anteroom.

The other two are: 1) trading in some of the objects from the showroom at the trading post (I know the ornate compass, not sure about other items) can get you an allowance token. 2) sometimes the experiment result from the lab will be to raise your allowance by one coin

Second method

You can carry your money (and gems) over to the next day if you find the freezer. Be aware that it will keep you from using money and gems the rest of your current day. Also if you go into the sauna or near the furnace after freezing it, your money can thaw out and you'd have to go back to the freezer to fix it.

If you haven't seen it yet, freezer is a dead-end room.

Third Method

The outside room the Tomb will generate five coins each time you create a dead-end room in your house.

Fourth Method

The office lets you spread coins through the house (or send them to the conference room if you've made one already.) This is 20 otherwise free coins. I know it lets you put pay slips in the Maid's Room and Servant Quarters too, but I have bad luck drafting them after doing so, so i'm not sure the result of that.

Fifth Method

Making certain tools in the workshop! Uh I almost forgot this and put another room first but this is WILDLY IMPORTANT.

Lucky Rabbit's Foot + Coin Pouch makes a tool that DOUBLES any money you pick up

the generic metal detector is good, but metal detector + shovel gets you an item that picks up more coins and keys from digging

Sixth Method

Various Laboratory experiments will get you cash. One of the possible results is to lose ten steps and gain 20 gold, which isn't necessarily something I'd recommend, but is an option. There are also results to randomly get a coin, key, or gem.

Seventh Method

The vault gives you 40 coins, just straight up.

Eighth Method

The Treasure Trove, which is a blueprint that you find, gives you five coins for each time you've drafted it, so the result is cumulative over time.

Ninth Method

Both the Observatory and the Laboratory experiment can generate chests in the entrance hall, which may or may not contain coins.

Tenth Method in tandem with 5

If you choose the root cellar as the outside room, you'll have more dig spots. It can add up.

Eleventh Method

SO

Once you have found the chess room (If you have found it you know what I mean, if you are not sure that you have done so you absolutely haven't) you can use it to set a condition to gain cash.

ETA: If you choose the blessing of the bishop, the chapel will give you 30 coins the first time you enter it in a day instead of taking a coin away each time you enter.

Getting to the chess room involves ...at least one puzzle just to get there, so I'll go into detail only if you ask.

Twelfth Method

AND SPEAKING OF THE CHAPEL

If you take something that can light fires to the chapel and light the candles with red stones in their holders, the altar will move up, revealing a piggy bank

If you smash the piggy bank, it will give back all the money the chapel took from you, though if you've been avoiding the chapel that might not be too much.

Thirteenth Method

Like the chapel, this is just...getting your own money back, but if you've been putting coins into the fountain outside, there's a way to lower the water level to get them back.

That's all I can think of off the top of my head; I'll add more if I think of it.

Oh this is all good to know!