Well I will try to explain how I have been imagining this, but please bear in mind that I am building on the ideas already suggested here so please don't think I am trying to claim they are all original.
Anakin - I like the idea of him being a space pilot of some description, and I would even have it so that he is carrying some sort of illegal cargo when he is first seen on screen. This way he can already have many bad habits and although he isn't evil it means it is easier to see why he fell to the dark side. He would also be a good contrast to Obi Wan because Obi Wan is the proper Jedi (so to speak). Where Obi Wan advises caution, Anakin rushes in (much in the same way he is already in the prequels).
Bail - He should be a Prince or King or whatever because then if we use the love triangle idea then that is where the title of Princess would pass from to Leia. I think Bail and NotPadme should have some sort of intimate history, but they are unmarried because NotPadme is not suitable and Bail is already arranged to be married to someone else. Then when NotPadme and Anakin hook up and Anakin turns to the dark side, Bail would immediatly offer to help NotPadme by looking after the children (showing that really he did love NotPadme).
NotPadme - Originally I was thinking she could be someone who was trained as a jedi, but left the order at a young age (for whatever reason). That way she would have some training but not be as good as someone like Obi Wan. Then somewhere along the line she ended up working as part of Bail's head of security or something like that, and thats how she met Bail. But someone made the point the jedi shouldn't be so organised as to be an order that you leave and join, and I actually agree with that too, so instead maybe she began training, but completed even less than Luke did. But there are 3 main reasons I think she should be a jedi:
1) It would explain why the force was so strong in Luke (because if Obi Wan trained Anakin and failed so badly then there would have to be a damn good reason for him to even consider training someone else).
2) I think if Anakin is a very bold and even somewhat reckless character, then the attraction to NotPadme could be more believable by her being a similar character. (Imagine when they first meet and he jokes about protecting her and really it is her protecting him).
3) She can be killed by Anakin in a lightsaber fight, and we will literally see that there is no good left in him, and it will give Obi Wan massive concern over the children's safety, to the point where he splits them up to hide them. And it would be a big enough shock to the system that Obi Wan would be so highly emotional when fighting Anakin that it seems like revenge for killing his friend, and this could be used in a 'will Obi Wan/won't Obi Wan turn to the dark side too?' kind of way.
Ihope I have explained that well enough, and I would love to hear what people think.