初めて details

for the first time — this occasion is the first of its kind for the subject + write

hajimete kaku

for the first time — this occasion is the first of its kind for the subject

meaning: write for the first time

明日あしたは、初めてはじめて会うあう人ひとばかりだ。

ashita wa, hajimete au hito bakari da.

Tomorrow it's all people I'll meet for the first time. neutral

これ、初はじめて食たべた。おいしいね。

kore, hajimete tabeta. oishii ne.

This is my first time eating this. It's good. casual

Examples

for the first time — the subject has never done or met this before

明日あしたは、初はじめて会あう人ひとばかりだ。

ashita wa, hajimete au hito bakari da.

Tomorrow it's all people I'll meet for the first time. neutral

これ、初はじめて食たべた。おいしいね。

kore, hajimete tabeta. oishii ne.

This is my first time eating this. It's good. casual

日本にほんに来きて、初はじめて雪ゆきを見みました。

nihon ni kite, hajimete yuki o mimashita.

I saw snow for the first time after coming to Japan. neutral

本日ほんじつは初はじめてお目めにかかります。どうぞよろしくお願ねがいいたします。

honjitsu wa hajimete ome ni kakarimasu. douzo yoroshiku o negai itashimasu.

This is the first time we have met. I look forward to working with you. formal

初めて and 始はじめて are homophones with different jobs: 勉強べんきょうを
始はじめて三年さんねん is "three years since I started", not "for the
first time".

たことがある
初はじめて marks the occasion as the first of its kind, 〜たことがある claims only that at least one such occasion exists — 初はじめて食たべた and 食たべたことがある divide "first" from "ever", and someone who has done a thing twice can say only the second.
まず
Both come out as "first" in English but count different things — まず orders steps within one occasion (まず手てを洗あらう), 初はじめて counts occasions across a lifetime (初はじめて食たべた), so neither can stand in for the other.

て初めて
〜てはじめて holds this very word but counts something else — it says the second thing became possible only AFTER the first (親おやになってはじめて分わかる), while bare 初はじめて just labels an occasion as the first, so the て in front is carrying the whole conditional reading.