"Which one will you drink?"
Assuming you know the kosoado system (many references online), どれ means 'which one', or simply 'which'.
を is a particle placed after a noun which denotes that it is being acted upon by the verb.
のみます means 'to drink', 'drink', 'will drink'. In this case, as it is a question not phrased in the past, the reasonable assumption is that it will take place in the future: 'will drink'.
か at the end shows that the phrase is a question.
The personal pronoun is dropped. It can be safely assumed that the omitted part is: あなたは
You / which one / will drink / ? or 'Which one will you drink?'.
The ability to use and understand the "topic marker" は is integral to proper understanding of Japanese grammar. While you can simplistically think of it as the equivalent of the English 'is', it has a more complicated function. は, が, に, を, へ, etc. all have important functions in making sense of the sometimes arbitrary order of verbs, nouns, noun phrases, adjectives, etc. and how they act with and upon each other. Without understanding them, one cannot truly understand the sentence. Ex.: Did John hit the ball or did the ball hit John? The particles will tell us that.
That is not to say that in colloquial conversation particles are not dropped. If context is clear enough, they often are, and this is what makes it casual and less precise. So, while the は is not required to be used in 100% of daily discourse, it will be expected that the function and usage be comprehended. Read some more about it here or among the many other resources for understanding Japanese particles available to you online.
Particle を shows the direct object. (i.e. みず を のみます。I drink water.) As opposed to が, which is used like は (wa) as a subject indicator. (i.e. いぬが のみます。The dog drinks.) Hope this helps!
Why do we have to use どれ when we ask for a drink and ごはん when we ask for food?
どれ is which or which one, and used both with のみます and 食べます (to drink or to eat). ごはん means rice or sometimes used to mean food.
It's teaching you how to say "I eat rice" or other variations and this one means "Which will you drink?"
When you get further along in the program you will run into どれを食べますか
I'm no expert, but when attached to the ます ending, む becomes み (changing the "u" sound to the "i" sound)