question about commas

Understand that commas are strictly a matter of style, and that opinions about them vary. So I can only tell you what I would myself do here. I would use commas in your sentence that this way:

1. I am travelling to New York, California, and Venice in July, and my friend Mary suggests I should also visit Rome.

You want the serial (“Oxford”) comma before the first and. There are several reasons for wanting it there, but one here is that you’re creating a complicated parse for the reader with all this bits that need commas, so you want to make sure they don't have reparse your sentence to understand it.

You want a comma before the second and because you’re joining two non-short independent clauses with a conjunction.

I would not surround Mary with commas because I take it to be a restrictive appositive. If you swapped the two and made it a nonrestrictive appositive or just made it a “heavier” phrase, then I would use the commas around Mary:

2. This July I’m travelling to New York, California, and Venice, and Mary, my friend from Italy, suggests that I should also visit Rome.

3. I’m travelling to New York, California, and Venice this July, and my friend from Italy, Mary, suggests that I should also visit Rome.

Or even with parentheses:

4. This July I’m travelling to New York, California, and Venice, and Mary (my friend from Italy) suggests that I should also visit Rome while I’m over there.