Usage of comma to join two sentences

A lot of times, I add a comma where I pause when speaking.

For instance, I might write:

"Thanks for speaking with me, I hope to see you again."

Someone said that this is not grammatical, and that I should either use a semicolon, or add an 'and' after the colon.

What are the grammar rules here?

This isn't really a question of grammar, but of punctuation.*

Formal English punctuation rules say that you don't join sentences with a comma. Sentences are separated with a period.

When you put a comma between two sentences, it's called a comma splice:

Comma splice

A comma splice occurs when two or more complete sentences are joined only with a comma, which is not strong enough punctuation. This is an example of a comma splice:

Joseph Cornell was an innovative American artist known for his shadow boxes and collages, his art became more famous after his death in 1972.

https://writingcenter.ashford.edu/run-ons-comma-splices

The two parts, "Thanks for speaking with me" and "I hope to see you again" are independent clauses so they cannot be linked with a comma. They could be connected with a semicolon, but this is a stylistic choice, not a necessity (and you'll find some style guides would tell you to avoid semicolons in most cases).

The basic idea (which you can see explained in better detail here: https://writingcenter.unc.edu/tips-and-tools/semi-colons-colons-and-dashes/) is that semicolons connect two clauses which are grammatically independent, but thematically linked. By using a semicolon between "Thanks for speaking with me" and "I hope to see you again" you suggest that your hope that you'll see the person again is connected to your being thankful to them.

But the semicolon doesn't suggest what the connection is, and a period doesn't imply that they're disconnected. Especially because, these days, periods can even be used (informally) in place of commas.

* The only way this could be a grammar issue is if you were somehow treating "I hope to see you again" as a dependent clause, which somehow modified "Thanks for speaking with me".