How would you say, I saw that movie for the first time? Would you use 初回 or a different word like 第一回? What are the differences between using these words to describe the "first time" doing something if there are any?
An adverb 初めて means "for the first time", and you should be using this in most situations. See this page for examples.
初回 is a relatively uncommon kango noun (see Sino-Japanese vocabulary and wago-and-kango). It's a stiff word used mainly in serious business articles or in technical/academic contexts. If you don't know how to use 初めて (JLPT N5 word), you may revisit 初回 after you have reached the N3 level. 第n回 is a way to count events that are held regularly (e.g. 第25回XYZコンテスト = "The 25th XYZ Contest").
I saw that movie for the first time. 私はあの映画を初めてみた。 for the first time = 初めて
Would you use 初回 or a different word like 第一回? No. 初回 and 第一回 means “the first time” ,Noun.