Learn Russian: Пациент идёт к специалисту. - The patient is walking to the specialist.

In Russian, when you are going to see a person (like a doctor, friend, or in this case, a specialist), you must use the preposition к. You use в or на when going to a physical place or an event.

Why is the word for specialist spelled специалисту with an -у at the end?

The preposition к always requires the noun following it to be in the Dative case. Since специалист is a masculine noun that normally ends in a consonant, we add the Dative ending -у.

Why use идёт instead of ходит for "is walking"?

Идёт is a one-way verb of motion, meaning the patient is currently making a single, specific trip directly to the specialist right now. If they went to the specialist regularly, you would use the multi-directional verb ходит.

How do I pronounce к специалисту? Is there a pause after к?

There is no pause! Single-letter prepositions like к are pronounced as if they are glued to the beginning of the next word. You should say it so it sounds like one long word.