Q: I want to give at least two weeks notice to my current employer and have at least 1 week off before I start my new position. How do I go about this and get what I want without jeopardizing the job offer?
A: This should be very manageable. Your negotiating window on a job offer is the time between when they propose and when you say "I do." (In marriage, the negotiating never stops.)
To negotiate your start date with a modest request like this, just lay it on the line:
"Oh, thanks so much! I’m really excited by this offer. It’s what I’ve wanted. I need to give my present employer two more weeks, and then I need another week or so to get there."
If you need 10 days, ask for 10. Don’t put an unreasonable demand on your moving time that means you’ll be asking for more time once you start. No employer wants to have a departing employee leave with no notice; no reasonable employer demands that a new hire have just a weekend between jobs.
Nine times out of 10, this kind of request is routine and handled without a blink.
Congratulations on the new job!