READ ME FIRST:
There are a lot of unclear points in this story, I will try to provide some clarity to it based on my understanding of the points made by the Mefarshim. Due to the length, I would like to post them as separate answers, and MY readers can vote on answers based on their content.
The 3 signs, as you mentioned, are explained in Shemos 4:2-9. Of those verses, two are of special importance to this conversation, verses 8 (commentaries here) and 9 (commentaries here).
4:8 - וְהָיָה֙ אִם־לֹ֣א יַאֲמִ֣ינוּ לָ֔ךְ וְלֹ֣א יִשְׁמְע֔וּ לְקֹ֖ל
הָאֹ֣ת הָרִאשׁ֑וֹן וְהֶֽאֱמִ֔ינוּ לְקֹ֖ל הָאֹ֥ת הָאַחֲרֽוֹן׃
The Lord said to Moses, If they do not believe you and are not
convinced by the first miraculous sign, they will be convinced by the
second sign (NLT)
4:9 - וְהָיָ֡ה אִם־לֹ֣א יַאֲמִ֡ינוּ גַּם֩ לִשְׁנֵ֨י הָאֹת֜וֹת
הָאֵ֗לֶּה וְלֹ֤א יִשְׁמְעוּן֙ לְקֹלֶ֔ךָ וְלָקַחְתָּ֙ מִמֵּימֵ֣י
הַיְאֹ֔ר וְשָׁפַכְתָּ֖ הַיַּבָּשָׁ֑ה וְהָי֤וּ הַמַּ֙יִם֙ אֲשֶׁ֣ר
תִּקַּ֣ח מִן־הַיְאֹ֔ר וְהָי֥וּ לְדָ֖ם בַּיַּבָּֽשֶׁת׃
And if they don’t believe you or listen to you even after these two
signs, then take some water from the Nile River and pour it out on the
dry ground. When you do, the water from the Nile will turn to blood on
the ground. (NLT)
Ibn Ezra's approach:
The first point that needs to be addressed is the following philosophical issue: Obviously, Hashem knows how many signs they will need in order to believe, so why bother telling Mosheh this? Just tell him to do the number of signs that it will take.
Ibn Ezra explains that God was saying this in order to reassure Mosheh - Hashem is basically saying "if you will be nervous that they will not accept you after just one sign, so perform the second and third signs":
אם לא יאמינו – ידענו כי השם ידע כל העתידות, רק הדיבור כנגד משה. אחר שאמרא לו השם: ושמעו לקולך (שמות ג׳:י״ח), אמר משה מפחד אני שלא ישמעו כולם בקולי.
ואם לא יאמינו – טעמו: אם תאמר שלא יאמינו, וכן: והיה אם לא יאמינו (שמות ד׳:ט׳).
He basically says that Hashem knows how many signs, but does not specify, because it doesn't matter. What is important is that technically speaking Mosheh might not know whether or not all of those present will accept it fully, and this is for Mosheh, so when it comes to performing the signs, he performs them all for the nation (likely to ease his own fears), as specifically stated by Ibn Ezra there:
...ויעש האותות – המטה, ואות היד, ולקח מים מהיאור והיה לדם ביבשה
Therefore, all three were performed, in order for Moshe to calm his mind. Ibn Ezra likely does not subscribe to this view of Chazal, as he does not mention it.