The ones with pearls (the pearls are the cost) will give you an additional slot permanently.

You make them in the milk machine, using milk from the milk plant

They add a temporary extra slot. Sometimes you get more of them from the villagers when you give them what they want.

They meant you send out your balloons to trade with other islands. When they come back, they bring a stone with them.

Just keep earning stars until you get to the number in the top left corner, and you'll move on to the next level.

Fountain upgrades all seem to require the same three types of elements (red, brown and turquoise). Will I need the green and blue ones in the future? I'm already at level 68 and haven't used one yet, and am wondering if I should dump them, as they are taking up storage space.