Learn Portuguese (Portugal): Eu bebo a água.

To say "I do something" (the Eu form) with regular verbs ending in -er, you drop the -er and add -o. For beber (to drink), take off -er to get the stem beb-, then add -o to make bebo.

Do I have to include Eu at the beginning of the sentence?

Not necessarily! Because the ending -o in bebo already tells us that "I" am doing the action, you can simply say Bebo a água. Including Eu just puts a little more emphasis on the fact that you are the one drinking.

In English, we often just say "I drink water" instead of "the water". Can I drop the a here?

Yes, but it changes the meaning slightly. If you want to talk about drinking water in general, you drop the article and say Eu bebo água. Because this sentence includes the feminine word for "the" (a), it specifically means "I drink the water" (referring to a specific glass or bottle of water).

Why is there an accent mark on the first letter of água?

The acute accent mark tells you exactly where to put the spoken stress. It signals that you should pronounce the word with the strongest beat on that first "a".