Learn Russian: Вы пьёте чай утром? - Do you drink tea in the morning?

The dictionary form (infinitive) is пить (to drink). It is highly irregular! In the present tense, the -ить drops off entirely, and we use a special stem with a soft sign: пь-. We then add the endings, but they start with ё or ю: я пью, ты пьёшь, он/она пьёт, мы пьём, вы пьёте, они пьют.

How do you pronounce the пьё part of пьёте?

The soft sign (ь) between the consonant and the vowel means you pronounce them distinctly, without blending them too much. It creates a slight 'y' sound separation, so it sounds like "p-YOH-tyeh" rather than a single smooth syllable.

Since "tea" is what is being drunk, why doesn't чай change its ending for the accusative case?

You are exactly right that direct objects usually take the accusative case. However, for masculine nouns that are non-living things (inanimate) like чай, the accusative form looks exactly the same as the basic dictionary form. It would only change its ending if it were a living person or animal.

The dictionary word for "morning" is утро. Why is it spelled утром in this sentence?