I am in a situation where I have been offered a job, with a proposed start date of 6 weeks away. So, is it appropriate to accept the job, and delay giving the two weeks notice to my current employer for about another 3 weeks? Is that a very bad idea? Would it be frowned upon? Will the new employer contact my current employer in any way in that time period?
Look at this way: if your company wanted to get rid of you they would have you pack up your stuff, and escort you off the premises before you could blink.
Right now you still have a job, and I see no reason to sacrifice your income on the altar of honor - most of us can't afford this luxury. Keep your head down, do your job to the best of your ability, and in 3 - 4 weeks, put in your notice. Nobody (and I do mean nobody) need know that you've already accepted another offer.
And do consider taking a week or so off between jobs. It's a great way to clear your head before you jump into a new workplace environment and the stress of a new job kicks in. Provided you can afford to do so, of course.
In the United States (since you did not specify your location), it is common and totally fine to give just 2 weeks notice, unless your employment agreement requires otherwise. Two weeks is perfectly professional and appropriate, and you may delay giving notice to provide just 2 weeks. You can tell your new manager when you are planning to tell your current manager, to make sure they do not contact them.
In other countries, these norms are different. In some places, employment contracts are common that require much more notice.
Let me also offer another perspective though. If you have a good relationship with your manager, and especially if you have a lot of projects or people under your authority, consider giving 3-4 weeks notice. It can be very helpful for your boss and yourself, to plan and make transitions. You do not owe this to anyone, since the standard is pretty universally 2 weeks, but I have been on both sides of this situation and it's a great thing to be able to do (if you have a good working relationship, and if you do not think it will jeopardize your employment).
A company will most certainly let you go at a moment's notice if they wanted you gone, so you really don't owe an employer anything. That being said, giving two weeks is usually a proper and courteous thing to do. There really is no rule (written or unwritten) that says that you have to tell your current employer that you have another job lined up that you're going to take in x amount of time or even that you're looking for another job. Most of the time when people give their two weeks notice it is understood that they have been looking and have already accepted a job by that point. Employers aren't dumb, they know in order to get to the point of giving two weeks notice that you've had to do a bunch of things over a good amount of time.
I don't know what field you're in, but in IT I've seen it go two ways when people give notice: 1. You're really busy for two weeks doing a knowledge transfer and tying up loose ends before you go. 2. You get your escort out of the building right after you give your two weeks notice. It all depends on the company and situation. Be prepared to leave your job when you give notice is all that I'm saying. Consider it a bonus if they actually let you stay for those two weeks.
As far as your new employer contacting your old employer - if you ask your new employer not to contact your current employer then they usually will abide by your wishes. Usually they understand that it could cause you issues.