Is the punctuation correct? Does it need another comma?

No, you need a comma after by the way.

By the way, Sharon, Robert and I are going to the river on Saturday.

This is still no good. From this sentence alone, it's impossible to tell who you're talking to. Are you telling Sharon that you and Robert are going to the river? Or are you telling someone else that you, Sharon, and Robert are going to the river? A serial com

Aspara GusFrom this sentence alone, it's impossible to tell who you're talking to. Are you telling Sharon that you and Robert are going to the river?

Hi Aspara Gus,
Thank you for your reply. Yes, I'm talking to Sharon.
Can I also use a semicolon instead of the em dash?
By the way, Sharon—Robert and I are going to the river on Saturday.

Well, an em dash isn't exactly the best choice, but a semicolon would be even worse because semicolons are used to link two independent clauses or to prevent confusion in lists. You might want to consider using one of my paraphrases.

Aspara GusYou might want to consider using one of my paraphrases.

Hi Aspara Gus,
Thank you for your reply.
aAccording to the context, I guess this is the one that, to me, fits better. Right?
Kate: By the way, Sharon, I'm going to the river with Robert on Saturday. Do you want to come with us?

I'm not sure what you mean by according to the context, but it is clear in both paraphrases that Sharon is being addressed, rather than talked about.

Aspara GusI'm not sure what you mean by according to the context, but it is clear in both paraphrases that Sharon is being addressed, rather than talked about.

Hi Aspara Gus,
Thank you for your reply. Sorry. The context that I know of, the way the story goes.

TS