Learn Russian: Я покупаю цветы.

The singular word for a flower is цветок. When making it plural, the -ок part drops out entirely, and we add the standard plural ending -ы to the base цвет-, resulting in цветы.

Since "flowers" is the direct object here, shouldn't it be in the Accusative case rather than the Nominative plural?

It actually is in the Accusative case! A handy rule in Russian is that for inanimate nouns (things that are not alive), the Accusative plural form is exactly the same as the Nominative plural form.

The translation says "am buying", but does покупаю also mean "I buy"?

Yes, it means both. Russian does not have a separate "am doing" continuous tense like English does. The present tense verb покупаю covers both "I buy" as a habit and "I am buying" right now.

How is the letter е pronounced in цветы?

Because the stress falls on the final syllable (-ы), the е is unstressed. In Russian, an unstressed е softens the consonant before it and sounds closer to a short English "i".