I really really like the unicorn I just started with. On other games I prefer to do a private lineage of my favorites and I was hoping to do that here. I adopted a LOVELY mare but didn't know that they would have previous owners horses and stuff. Is there any way to do a private lineage here? I know it might sound dumb but I just find it fun. 

I think you will want to catch wild pets. Under the Pet- tab you can find ^Capture Wild Pet^. I haven't used it much, so I don't know how you do it the best way. Look for a players guide for further information, they are really helpful.^^

The only issue with wild pets is getting good genes - it's really a crap shoot. However it's great for breeding projects as there is no possibility of inbreeding. My pets in some lineages are so closely related it's more of a family twig than a family tree.