I do home daycare and our tax laws are a bit different since we do it all in our house and we feed other kids and stuff like that, but I can give you some pointers.
First the first year you won't really be able to pay quarterly, the penalty for not doing so will not be anything this year because he is basically starting at the start of the last quarter of the year. So, after you file those taxes you will have a better idea of what his business tax amount will be and can either file quarterly if indicated (Turbo Tax tells you how much and how to do it and everything if you have to do this).
Save every receipt, and get a receipt, for everything for this coming up quarter. Have him write down mileage, or keep a log of where he went at least so you can googlemap the distances later, each time he has to use his car for anything work related (even picking up office supplies for use at home would count). It sometimes helps if that car becomes the car for him to use for business only, but if that is not possible it may also be possible to just add up all the OTHER stuff he does in the car (library every Wednesday night is 3 miles, driving daughter to school is one mile, etc.).
Anything he buys have him put a receipt in a basket when he comes home. Hang one in a very obviously unavoidable place. Tell him he can't make a purchase at all, even McD's, without getting a receipt from now on. When it comes January 1st take all those receipts and put them in an envelope and have him start a new pile. I would personally do his portion of the taxes as soon as you can. Either find a tax person who has experience with independent contractors (and ask them for references) or do it on Turbo Tax. Turbo Tax walks you through it every step of the way and is totally user friendly for this, just make sure you get the home business program.
It is wise to put away 30% this quarter. If you do your taxes soon in the next year you will have a better idea of what number you really will need.