You can use either of these sentences. It's true that the "go to" can be omitted, but this omission isn't specific to the verb go. All the following are used:
I'm going to London on 1st October.
I'm visiting London on 1st October.
I'm seeing David on 1st October.
I'm meeting Marie on Tuesday.
I'm having dinner with Brian next week.
(The "going to go to", "going to visit" etc forms would be equally correct. "I am" and "I'm" are interchangeable, but "I'm" is far more common in speech and in writing to friends and colleagues.)
You generally have to include the time expression (such as 1st October or next week) - otherwise it implies that you are visiting London right now or meeting Marie imminently or in the process of having dinner.
However, the time expression could be omitted if the future time was implied by the context. E.g. "I can't be there tomorrow night." / "Why not?" / "I'm meeting Marie."
Similarly, if you talking to someone in person (or in another situation where they can be expected to understand that you're not currently in London), "I'm going to London" could be shorthand or "I'm going to go to London". Your friend will probably then ask you "When?".
In cases where there is no time expression and where the context doesn't make it clear that you're talking about the future, you must include "going to":
I'm going to go to London.
I'm going to visit Paris.
I'm going to have dinner with Brian.
Note on dates:
- (BrE): Except in legal documents, people hardly ever write "the 1st of October". We write "1st October" or (less commonly and less conventionally) "the 1st October" - but we pronounce it "the 1st of October". The words "the" and "of" are mandatory in speech but rare in writing. Similarly, "October 1st" is pronounced "October the first" and has the same meaning. (Note, if the month is obvious from the context we can also say "on the 1st". Here the "the" will be probably be written down.)
- (AmE): The form "October 1st" is strongly preferred in most contexts, and pronouncing the word "the" is optional. When referring to the Independence Day holiday, the form "the Fourth of July" is often used and written in full.