The three phrases are just different ways of saying the same things, just some are more formal than others.
I have to go - Casual but much more formal than I gotta go.
I need to go - Same as above
I gotta go - something I would say to my friends on chat or if I meet them in the street and need to leave quickly.
Corrections to your question:
"I have to go" vs "I need to go" vs "I gotta go". Do these three sentences have the same meaning?
I usually watch some series and it´s very common to hear: "I gotta go". However, I think that these three expressions have the same meaning. Is that true or not?
Ps.: I would be glad if you guys also could help me by correcting my mistakes in my question. Thanks in advance.
PS. Your english is great!