I'm not 100% sure how I feel about the rainbow colors. I think I like them when considering only the logo... but I'm having trouble visualizing a full website that would match this design (which is more a fault of my own than with your design).
I like Entry #2 more than #1. However, in both cases, if the logo is going to have a picture in it, I think it should be a pacifier (but I'm open to other ideas). I'm just not a huge fan of the Sun, I think. Mostly because I don't want my primary color to be yellow and also because I think it's very common for Daycares to use Sun images.
Overall, Excellent work! I'm very excited to see more.