Logo Help, Need help with a company logo Options

[Datrooper]

Ok, well I'm not very good with logos to start off, I don't really have any creative ideas. The red bird named "einstein" is what the company is using now. They don't like it, because it's too sharp, they'd like it to be smoother. However, they kinda want a different animal, and they told me that like the idea of an owl reading a book. So, i kinda have that as inspiration i guess. They like the vector logo, and they like how it is cartoonish, since the company is about parents reading to their children. So, could someone help me out? I'm not really asking for someone to do it for me, but i'd like someone to kinda point me in the right direction. Like, I don't know how to make logos at all. I have photoshop, and illustrator, so i think i have the right tools. Like, what are some steps to designing a logo? Also, i'll put a picture of a logo that they like.

Ok, here are the images:

click to enlarge

click to enlarge

Thanks in advance for you advice!

-Luke

EDIT: I'm trying to redraw that owl logo in illustrator, but making my own changes, if it is distinctively different, will it still be copying? And can the company use it?