I often see sentences where the person would say "my first initial reaction" and my question is should first and initial be used together? I don't like the sound of it - I would say either. my first reaction or my initial reaction - I don't know why this bothers me so much but there it is.
These are adjectives, not adverbs, but the question remains.
The two adjectives are redundant. They have the same meaning; you don't need both. In informal writing or casual speech, this is reasonably common, usually to emphasize the adjective more. But in formal writing, you would want to avoid including both words.
Not inherently wrong, but you should avoid it.
For a not-wrong example: "When my dog died, I had two initial reactions, and a third reaction which came only after a long time. My first initial reaction was rage. Rage against the world! Second, I also felt sadness. Sadness like an ocean of tears. Etc."
"Initial" refers to a beginning. It is valid to describe multiple things happening during a beginning.
However! If you start with "first," that implies that there is a "second" coming, and likely a "third." Unless you are going to list out a sequence, don't say "first."