The currency in Meet-me is the ココア (Cocore), often shortened in game to 'Co'.
When a Meet-me account is created, 2000Co are awarded automatically.
This is displayed in the green bar at the top of the screen.
The blue bar represents a secondary currency called Medals.

One way of obtaining Cocore is to take items to a government office (yakusho 役所). There are numerous government offices around Tokyo, all of which can be reached by bus stop. Look for the kanji 役所前, which means "in front of government office"
One government office is found at the bus stop directly east of the Tokyo Tower and is visible from the tower.
It is possible to get here directly from the bus stop north of Roppongi 六本木.
This is the kanji for this office's bus stop
Another government office is reached by bus from outside the starter apartment block and is in west Tokyo.
The kanji for its bus stop is

After going into the government office, a machine to the right hand side is visible (looks like a cashpoint / ATM). This machine converts items into Cocore.
The size and rarity of the items seems to be related to the amount of Cocore received. Most items from gatcha-gatcha machines are traded for between 25Co - 80Co.
Co is used to buy food, items and stamps for sending letters as well as pay for ground rent.
To earn Co, billboards can be clicked on which ask a multi-choice question. If the question is answered correctly within the time limit, 20Co are awarded.
Medals are used in Medal shops.
No way of selling items has been implemented yet, apart from at the Government offices.