If I do this though, I would want my own personal page to not be linked or affiliated with the daycare fb page at all! It just seems more professional IMO. (not that I think you are not professional cfred!!) I am just not sure how to go about doing that. Fb settings seem to magically change themselves sometimes lol.
Unfortunately, the way FB is set up, only a few scenarios will suit you.

A FB page is public - you have no control over who can view it. This can have a completely separate name and not be linked to you personally but all it's good for (IMO) with this business, is generic information such as business hours, little tidbits about activities and photographs with do not show the children's faces. Think of it like a one page website. All content is available for all to see.

The alternative is FB groups - there are several options here. A public group where anyone can add themselves and see all the content before they do so. A group with members, where again, the content is publicly visible but an Admin agreement is required before a new member can join and comment/add. Or is private (secret) group, which is not publicly visible, cannot be found be searching, where the Admin (you) has to add members and can only do so, if they happen to be FB friends with that member. With regards to all the groups, a search within the group will show your personal account as the Admin. The publicly seen groups will only permit access to your information in as much as what is already visible to a non-FB friend who looks at your profile. (as per your own security settings). A private group will be allowed to view your personal page based on the need for them to be FB friends in order to be included in the group - however, again in your own security settings, you can limit individual's on your friend list to only see what a non-member does by making them a restricted friend. This is how I get around it.

My day care clients cannot see any more of my personal FB account, than a non-friend member of FB can. I am not on the public search. If I need them to see something I am posting on my account, I would have to change the settings for that post to public. They can however see everything that is posted in the private day care group pages.

I don't mix business with pleasure. I don't aim to be a friend of day care clients. There is no reason at all, for these people to have access to my family and friends. They don't need to see what my great Aunt wrote on my wall for my birthday, they don't even need to know it's my birthday.