Learn Russian: Я читаю стихи.

Because стихи is an inanimate noun (a thing, not a living person or animal). In Russian, inanimate nouns in the Accusative case look exactly the same as their Nominative (dictionary) form. Direct object endings mostly change for singular feminine words or living things.

Is стихи a plural word? The translation says poetry which is singular in English.

Yes, стихи is the plural form of стих (a poem or verse). While English uses the uncountable singular word poetry, Russian commonly uses the plural стихи to express the exact same idea. And since it is an inanimate plural, it does not change its ending when used as a direct object.

How do I know if this means I read poetry or I am reading poetry?

You just have to rely on the context! Russian does not have a separate present continuous tense for actions happening right this second. The verb читаю covers both I read (as a general hobby) and I am reading (right now).

If I wanted to say I read a poem (singular), how would that look?

The singular word for a poem is стих. Since it is an inanimate masculine noun, its Accusative form is also identical to its dictionary form. So, I read a poem is simply Я читаю стих.