I've done this enough to know, in most cases you if you are up front, give them as much notice as you can, it should not be a problem.
Leave out the unhappy part, none of that is relevant now, whether its true, partly true, or just on your mind. You can state in a polite manner that you have a chance to better yourself with the new job, no need to give details, you never know if they will want to barter with you, or even with 2-4 weeks notice, "we don't need you anymore" and you are terminated on the spot. Most are gracious when you give adequate notice, some are ok with it, some may not be, its a gamble, but best you do it honorably in good faith on your end always. The object is to never burn bridges, something happens with the new gig, you can go back, I've been told that many times, we can't match the new job, but if for some reason it does not work out, you are more than welcome to come back. You'll have to manage that week off, bad time for that to happen and give notice, but it has nothing to do with the new job, its family and you need the time, I'd tell them your notice is 2 weeks after that week off, so make it 3,. Those who have hired you with the new offer, should never pressure you to burn a bridge, I've always been firm on that and I've had at least one really pressure me to leave sooner and go back on my word to the previous employer, something I do not bend on, always best to do this way in advance on both ends, new employer should realize that if you short them, someday you will do the same to them. Most times it is they that are in a bind and try to make it up by getting you in sooner or what have you, its not fair to you or the previous employer, just don't wait til the last minute and say I was also unhappy, then also have to take care of a relative immediately for week, but I'll be back the following week, at that point you may just be asked to leave anyway, bridge burnt etc.