"And go on" isn't correct.
As for "etc." vs. "and so on", they can be used interchangeably, BUT:
1. "etc." is an abbreviation, so it should always be written with a period at the end, even in the middle of a sentence. In Spoken English we use the full term "et cetera".
2. These aren't typically used to refer to lists of people, especially not "etc." because it literally means "and other things"
3. "Etc." should only follow lists of 2 or more items, but "and so on" is sometimes used after only one example.
You could say "My mother does the cleaning and so on."; meaning that she cleans, and also does other related household tasks (cooking, laundry, etc.).
Besides these, you can also say "and so forth", or combine them into one term, "and so on, and so forth". The latter isn't that common; when I hear people use "and so on and so forth", it can imply that the list is too long, or that the items in list that aren't mentioned are unimportant.
"And whatnot" is something you'll hear sometimes, although it sounds very colloquial/informal.
"And the like" is similar, and more suitable for lists of people ("Pop stars, movie stars, models, and the like").
