What happens if a land is granted two different basic land types — can it tap for both colors of mana?

Each of the five basic land types — Plains, Island, Swamp, Mountain, and Forest — intrinsically grants a mana ability to any land that has that type, regardless of how the land acquired it. This is established in CR 305.6: a land with the subtype Plains has the ability "{T}: Add {W}", Island gives {U}, Swamp gives {B}, Mountain gives {R}, and Forest gives {G}.

When a land is granted two different basic land types (for example, by an effect that says "it becomes a Plains and an Island"), it gains both associated mana abilities simultaneously. The land can tap for either color independently — you choose which ability to activate each time you tap it, but it can only be tapped once per use (unless untapped again). This is a natural result of CR 112.1 and 305.6 working together.

Importantly, if an effect simply grants the subtypes without removing existing ones, the land may retain other abilities too. But if the effect says the land "becomes" those types, it typically loses any previous basic land subtypes it had (CR 205.1b, 612.3).

Concrete example: Your opponent plays Spreading Seas on your Stomping Ground, making it "an Island in addition to its other types." Your Stomping Ground now has both Mountain and Island subtypes. You can tap it for {R} (from Mountain) or tap it for {U} (from Island) — two separate abilities, your choice each time you tap it.

So the short answer: yes, a land with two basic land types gains both mana abilities and can tap for either corresponding color of mana, one at a time per tap activation.