Learn Russian: Вчера сильно шёл дождь.

You would change the adverb сильно (heavily/strongly) to its comparative form сильнее, and add an intensifier like гораздо or намного. For example: Вчера гораздо сильнее шёл дождь or Вчера дождь шёл намного сильнее.

Why do we use the verb шёл (went) to talk about rain?

In Russian, precipitation "goes." You use the verb идти (to go on foot) for weather events like rain (дождь) and snow (снег). Because дождь is a masculine noun, it takes the masculine past tense form шёл.

Why is the adverb сильно (strongly) used instead of a word that literally translates to "heavily"?

In English, we say rain is "heavy," but Russian expresses this concept with the word for "strong." You use the adverb сильно (strongly) to describe the action of the rain falling, or the adjective сильный (strong) if you are describing the noun itself, as in сильный дождь (heavy rain).

Is it normal to put the subject дождь at the very end of the sentence?

Yes, Russian word order is very flexible and uses the end of the sentence to introduce new information or set a scene. Вчера сильно шёл дождь sounds very natural. If you said Вчера дождь шёл сильно, the focus shifts slightly to how the rain fell (heavily, rather than lightly).