Buildings can be upgraded by using materials found in Star Boxes. Materials are also sometimes awarded as bonuses when you play Hidden Object Scenes.
When you click/tap on a building, some pop-up buttons and info will appear at the bottom of the screen. This shows some information, like the building's name, the building's current level, and the number of Flowers it is currently worth.
Tap on the green Upgrade button to open the building's upgrade screen. You will then see the materials required to start the upgrade process, and the time it will take for the upgrade to finish.
You can instantly upgrade the building with Diamonds. To do so, tap 'Instant Upgrade' to buy any missing materials and also skip the construction timer so the upgrade happens right away!