First of all (yes, how do we say that?) let's start with the simple count.

How do we say "the first day" (which means Sunday, by the way 😉)? Right, יום ראשון

So, how do we say: "What was the name of the first man on the moon?" You say:אֵיךְ קָרְאוּ לְבֵן אָדָם הָרִאשׁוֹן עַל הַיָּרֵחַ?

More examples:
The first thing I wanted to tell him when I saw him was …הַדָּבָר הָרִאשׁוֹן שֶׁרָצִיתִי לְהַגִּיד לוֹ הָיָה …
What - I am the first one to come? Unbelievable!
מָה זֶה - הִגַּעְתִּי רִאשׁוֹן? לֹא יֵאָמֵן!

So this one is relatively simple – and exhausts the usage of the word ראשון
I mean – use it when you count people, things or objects, and this is it.

Then, let's look at "AT FIRST. "
We are counting nothing here, right?
So – "rishon" cannot be used for saying "AT first", never! 😉 What you want to say here is "at the first moment," and our intention is to describe the state of things when you start your observation, as opposed to what you saw next. "Starting" is "beginning." So, in Hebrew, what we actually say here is "in the beginning" – בהתחלה.
Examples:בְּהַתְחָלָה חָשַׁבְתִּי שֶׁאַף אֶחָד לֹא נִמְצָא, אֲבָל אַחַר כָּךְ הֵבַנְתִּי שֶׁפָּשׁוּט כֻּלָּם הָיוּ בְּחֶדֶר אַחֵר.
At first I though that there is no one there, but then I realized that they have just been in another room.
בְּהַתְחָלָה זֶה יָכֹל לִהְיוֹת קָשֶׁה לִקְרֹא מִיָּמִין לִשְׂמֹאל, אֲבָל אַחַר כָּךְ מִתְרַגְּלִים. 😊At first it may be difficult to read from right to left, but then you get used to it.

"First, I want to tell you that …. Second, … Third, …" What do we do here?! Is it – counting? So it's the first case? But there is a comma, so – is it the second?! Ahh, it's impossible! No panic! Yes, it's both, and since it's both, there is an ultimately elegant solution!

First, (קֹדֶם כֹּל) I want you to think – how would you say "my son is in the first grade now" – which word would you use for "first" here, eh? 😉
(Here is a song for you as a hint! ;) )
Did you guess it? Right, it's just grade "A"- "kita alef" – כיתה א' Which means that we can count with letters, did you know that? ;) ***
And this brings us to the solution for the case when you want to count the possibilities "first, …, second, …, third, …" You just say "ALEF, …, BEIT, …, GIMEL, …"! This simple! :D

Example:אָלֶף, - אֲנִי לֹא יוֹדֵעַ אֵיפֹה לְחַפֵּשׂ, בֵּית, - זֹאת לֹא הָאַחְרָיוּת שֶׁלִּי, גִּימֶל, - אַף אֶחָד אֲפִלּוּ לֹא בִּקֵּשׁ מִמֶּנִּי לַעֲשׂוֹת אֶת זֶה! First, I don't know where to look for it, second, it's not my responsibility, third, no one even asked me to do it!

***
Counting with Hebrew letters has a name to it – how is it called? If you know – write back to me with an answer! 😉