I see you have gotten a LOT of responses, and I really don't have the time right now to read through them all, so I am just going to offer you my 2 cents, and my apologies to anyone who may have already told you this.
This is VERY common for the first year of school, preschool, or daycare. It happened to all my kids ( in school ), my grandkids (in daycare), to me when I was teaching my first year at the grade school, and my sister her first year working at the pediatrician's office.
The pediatricians in my sister's office called it the "First Year Crud". They said they all had the problem, too, when they first started working with children. You get exposed to all these fun little germies that children so generously pass around, and it takes your first year to build up immunities to them. The pediatricians actually said it was a good thing. (yea, right .... I dare them to say that to a mom who is cleaning up a puking child and all his/her bedding for the fourth time in one night) They said it gets those little immune systems built up.
Just try to keep your child up to date on their immunizations, feed them well, and be sure they get plenty of rest. Then you just have to doctor the various "cruds" as you get them ( there are things out there I had never even heard of before ! LOL !) and even though it seems never ending, it will get better in time.
Incidentally, you and the rest of the family will get most of this stuff, also, as your sweet child generously shares all their interesting new germies with you at home.
Ain't it fun ?
LOL. Sorry. That probably isn't funny to you right now.
I really do feel sorry for you.
Been there, done that, bought the T-shirt ........... SO many times !
Hang in there. It does get better.