The transition that best indicates a chronological sequence is "First, I needed to act as if the ridicule didn’t bother me at all.", chronological events are easily and commonly organized with the use of the following words: FIrst, Secondly, Then, After, Next, FInally, etc...
The other sentences present different structures.
"My face instantly flushed red, and I knew that I had to formulate a plan." uses a time adverbs but it is not a sequence one.
"Yesterday, I went to lunch as if nothing had happened." Yesterday is a time expression that doesn't necessarily demonstrates the order of events.
"On the one hand, they did not care that I was embarrassed." On the other hand, is used to introduce a contrast.
"Fortunately, they did not notice, and I could pretend not to care." This sentence uses an attitude adverb.
