Learn Danish: Jeg taler både dansk og engelsk. - I speak both Danish and English.

In Danish, names of languages and nationalities are written with lowercase letters, unlike in English. So it is dansk and engelsk, not Dansk and Engelsk.

How does the phrase både ... og work?

Både ... og translates directly to "both ... and". You use it exactly like in English to link two things that are both true. For example, både dansk og engelsk means "both Danish and English".

Could I use begge instead of både to mean "both" in this sentence?

No, begge is used when you are just saying "both" or "both of them" (like "I speak both"). But when you explicitly list the two things separated by "and" (X and Y), you must use the set pair både ... og.

What is the difference between taler and snakker? Could I use snakker here?

Both mean "speak" or "talk". Taler is the standard, slightly more formal word used when talking about the ability to speak a language. Snakker means "chat" or "talk" in a more casual way. While you might hear Danes say jeg snakker dansk, taler is the safest and most natural choice for languages.