Ask at any place if they will allow you to put them up. As for putting about the neighborhood, put them either on folks doors or ON the mailbox with tape. Do not put them in the mailbox, that s illegal, as per the post office.
the best place is to post ads on http://www.craigslist.org for childcare. also post in the community sections. it will give you alot more traffic than flyers. plus doesnt cost a cent no print ink or paper costsbut yes ask for permission when placing flyers.