Today's shaping up to be a good day.
I looked up what shaping up means. I think it means, the sentence, today's turning into a good day. Am I right?
And, does today's shaping into a good day mean the same thing?
"To shape up" is informal for "to develop" or "to turn into":
It carries a nuance of "in my opinion" and/or "contrary to my expectation."
"Shape up" is a phrasal verb; you can't drop the "up." So your second sentence is ungrammatical. Another common way to say the same thing is:
Today's looking like it's going to be a good day.
With many variations.