How do you upgrade towers in Bloons Tower Defense?

You need to unlock the higher level versions of the towers in the city portion of the game. Each tower has a building that holds upgrades and a building that lets you deploy more copies of the tower during battles. There are also some late game buildings that need to be built before you can purchase and research late game upgrades. Every tier 4 upgrade will require one of these special buildings. A few of the tier 3 upgrades on advanced towers will also need these special buildings.

Upgrades are purchased at the upgrade building for a certain amount of coins and gems. After the purchase, they will take some amount of real life time to be researched (a few minutes for tier 1 upgrades, hours and eventually days for the higher tiers) then you can click/tap to complete the research and finally they will be unlocked for use during battles.

The missions and tutorial during the city portion should cover this as you progress through the game.