I'm a British English speaker. There may be differences here between British and American usage.
"I have to [do something]" is fine.
"I've got to [do something]" is also OK, but is more informal (the contraction "I've" would very often be used). This use of "got" is extremely common in everyday speech and writing, but I would avoid it in formal writing.
In certain parts of Britain (e.g. Scotland, but maybe others), "I got to go" has an entirely different meaning: "I was permitted to go" or "I was given the opportunity to go". Similarly "Did you get to go?" means "were you given the opportunity/permission to go?"; I have even heard this abbrevated to "He'll no' get" meaning "he will not be able to go". etc.