There are tons of cats that you could collect in the game and a few that you need to pay actual money for which makes it a little frustrating for F2P achievers-- people who collects game achievements for free.
A little warning is the amount of grinding and patience you need to have for this game. There's a daily login attendance and in completion of the week, you get a Mochicat and it comes in random in every first day of the 7-day login-- a Mochicat you haven't collected or the ones you already have-- rarity comes in random and I had 3 S rarity Mochicat in 3 separate 7-day logins.
Mochicats available in the SHOP are the ones obtainable as free in the game.
There's the DESSERT TOWER (that's what the Incomplete English says which the feature was removed in the later updates) which are sort of stage necessary to get a Mochicat of the dessert tower's theme (ex: Fast-food Tower gives Fast-food Mochicat) and everytime you unlock a tower, you must build it with the MOCHI currency. Each tower has 5 Mochicats available and the PARTY feature works like gachas that needs PARTY TICKETs but a lot assuring. There's a CHALLENGE each tower which requires 6 mochicats in your team-- 5 of the Mochicats are the acquirable Mochicat in each tower and it's permanent, which means you can only add 1 Mochicat for each tower.
Certain events are available monthly in the game and during those events, the rewards in the WHEEL OF FORTUNE (the harbinger of gacha frustrations) differ, with two Mochicats available which you can't get in any way other than WOF. There are also this daily and weekly challenges during the events which allows you to gather points and the more points you get, the more rewards you can obtain. Some events reward you with a Mochicat (only available this way) as a top reward, and the game mails you a gift if you participated the event after it ended, and I highly recommend you to do the event if you're like me (who has a goal to get the available rewards).