Noticed a cold sore coming on, what can I do to stop it from coming on fully? Or to shorten its stay?

Inside your mouth? I used to put an aspirin between it and my gums, and it went away. I have no idea if it was the dry pill, the aspirin ingredient, or if it was about to go away on it's own.
Any drugstore will have remedies too.
A cold sore outside is best treated with an OTC topical product. Keep it clean and dry, and don't irritate or rub it.

I had the best luck getting rid of a cold sore on the outside of my lip when I held an ice cube against the tingly spot (first sign of one). I did an on-for-five-minutes, off-for-five-minutes thing. If one popped up without the ice cube treatment, I held a cotton ball soaked in rubbing alcohol against it several times (and screamed so the neighbors probably heard me). I haven't had one for years, but my husband gets them when he eats tomatoes or is out in the sun too long, but treats them with a topical ointment (I'll ask him the name of it) once they have appeared. He used to take lysine to prevent them (and it worked!), but stopped doing that for some reason.