Learn Portuguese (Portugal): Eu vejo o médico três vezes.

The verb ver (to see) is irregular. If it followed the normal rules for verbs ending in -er, the "I" form would be "veo". Instead, it gets an extra "j" to become vejo.

How do I know when to use vezes instead of tempo for the word "times"?

Use vezes (the plural of vez) when you mean "times" as in countable occurrences or instances, like "three times". Use tempo when you mean time as a concept (like "I don't have time") or the weather.

Why do we need the article o before médico? Don't we drop the article for professions?

We drop the article when stating someone's profession, like Ele é médico (He is a doctor). However, in this sentence, you are interacting with a specific person ("the doctor"), so you must use the definite article o.

Can I drop the word Eu from the beginning of this sentence?

Yes! In Portuguese, the verb ending usually tells you who is doing the action. Since vejo can only mean "I see", you can safely drop the pronoun and just say Vejo o médico três vezes.