One day, as she did any other day, my mother said to me that it was time to go to the bathe. One day, as she did any other day, my mother said to me that it was time to bathe.

we don't say "go to bath"

One day, as she did any other day, my mother said to me that it was time to go to the bathbath time. One day, as she did any other day, my mother said to me that it was bath time.

In most contexts as an adult, you would say "to take a bath." But in the specific context of talking to children, it's more natural to say "it's bath time." (At least in the US.)

Instead of waiting for her help, I took myself my clothes off myself. Instead of waiting for her help, I took my clothes off myself.

usually "myself" will go at the end of the sentence, there are a few exceptions as it changes the emphasis but the most standard would be at the end

Instead of waiting for her help, I took myself my clothes off myself. Instead of waiting for her help, I took my clothes off myself.

But I forgot to remove my socks. But I forgot to remove my socks.

some people say there's a rule not to start sentences with "But". it's not completely strict but it would probably be better to connect this sentence with the previous

But I forgot to removetake off my socks. But I forgot to take off my socks.

It's possible to say "to remove one's socks" but it's oddly formal. It's more natural to say "to take off one's socks."

Before my mother could stop me, I climbed enthusiastically on the stepstool thenand jumped into the bathtub. Before my mother could stop me, I climbed enthusiastically on the stepstool and jumped into the bathtub.

Old enough to undress myself and enter alone into the tub. Old enough to undress myself and enter alone into the tub.

technically not a complete sentence but otherwise fine

Old enough to undress myself and enter aloneget into the tub all by myself. Old enough to undress myself and get in the tub all by myself.

You would always say "get in the tub" and never "enter the tub." I wrote here "all by myself;" adding the "all" adds special emphasis in the way a kid who feels proud might, or an adult looking back on their childhood and voicing the perspective of their younger self. An adult speaking as their adult self would probably just say "by myself" or "on my own."

Young enough to forget about the socks and also to make of something that insignificant into a long-lasting memory. Young enough to forget about the socks and also to make something that insignificant into a long-lasting memory.

I think that only a child can think that wet socks are such a big event, that it is worth remembering! I think that only a child can think that wet socks are such a big event that it is worth remembering!

you don't use commas before relative clauses with "that"