Завтра я пойду к парикмахеру после работы.

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Breakdown of Завтра я пойду к парикмахеру после работы.

• пойти: to go
• к: to
• завтра: tomorrow

Questions & Answers about Завтра я пойду к парикмахеру после работы.

Why does пойду mean “I will go” even though it looks like a present-tense form?

In Russian, many perfective verbs don’t have a separate future form. Their “present” endings are used with future meaning.
Пойду is from the perfective verb пойти (“to go (once), to set off”), so я пойду = “I will go / I’m going to go (once).”
If you used an imperfective verb, you’d typically form the future with буду + infinitive, e.g. я буду идти (“I will be going” / focusing on the process).

What nuance does пойду have compared to буду идти or пойду ходить?

• (завтра) я пойду: one конкретное trip/action (“tomorrow I’ll go (there)”). Most natural here.
• (завтра) я буду идти: emphasizes the process of walking/going (“tomorrow I’ll be on my way / I’ll be walking”). Often used when describing what will be happening at a certain time.
• я пойду ходить is generally not what you want here; ходить is “to go regularly / back and forth.” For habitual future you’d say something like я буду ходить к парикмахеру чаще (“I’ll go to the hairdresser more often”).

Why is it к парикмахеру? What case is that?

The preposition к (“to, toward”) requires the dative case.
парикмахер (hairdresser/barber) → dative singular парикмахеру.
So к парикмахеру = “to the hairdresser” (i.e., to that person / to their place).