From a manager's point of view (the manager you're departing from), it's hard to find good replacements, and you always wish you could keep your best people longer, but so long as I have appropriate notice, then the team around you should be able to prepare to take up the slack. Sure, it's going to take weeks, maybe months to train a replacement, but that's not your concern.
I'm sure it would be appreciated by your previous employer if you stayed longer (if they like you, I'm sure they'd prefer you actually didn't leave at all), but so long as you're professional, courteous, and give sufficient notice (I think up to four weeks is generous, unless your a high-level execute, in which case longer departures might be more normal), your previous company should be fine with it. Changes in employees are a normal part of the process. I'm not saying that everyone is infinitely replaceable, but no company can survive if it's not able to change its staff.
If you really want to help them, you can possibly offer to make yourself available via email (outside your new employer's business hours, of course), but that's really going over and above, and not at all usual or customary. Not only that, it's actually just kind of weird, and I would recommend against it.
Whatever happens, figure out an official start date for the new place, and stick to it. If circumstances at your previous employer change at the last minute, and they really want you to stay an extra couple of weeks, well, that's really the previous employer's issue, I'm afraid. You've made a commitment to the new place, and you absolutely need to keep it. It's important.
Finally, to put it one last way, you're leaving one relationship, and going on to another. Clean, courteous breaks tend to be the best. You wouldn't make yourself available for an unknown amount of time to a previous boyfriend/girlfriend, would you? If the person you were leaving wanted you to "stay available" (my words) for a few weeks afterward, wouldn't you think that wasn't exactly healthy, and that they should deal with the reality at hand, instead of trying to hold on to you?