Learn Spanish (Latin America): Voy a la playa.

You would use adónde (to where). Because voy a la playa talks about moving toward a destination, the matching question word needs the 'to' built in. The full question would be ¿Adónde vas?

Why isn't the word yo (I) at the beginning of the sentence?

The verb voy already means 'I go' or 'I am going.' In Spanish, since the verb changes shape depending on who is doing the action, you can usually drop words like yo (I) or tú (you) unless you really want to emphasize who is doing it.

Why don't a and la combine into a smaller word here?

You might be thinking of the rule where a (to) and the masculine word el (the) combine to make al. However, playa is feminine, so it uses la. The words a and la always stay separate, giving us a la playa.

The translation says 'I am going,' but isn't voy just 'I go'?

It is both! In Spanish, the simple present tense like voy is used for general habits ('I go to the beach every summer') as well as actions happening right now ('I am going to the beach'). You don't always need a special '-ing' verb form to say what you are doing right now.