Try and make an image of how you want things and it'd be easier to help from there.
I believe several of us are using a modified version of code we found
over here. In that thread we changed it so rather than displaying pet information in a row we display each pet it's own cell, side by side. From there it's much easier to style the table without needing to remove it.
Inevitably, styling is always done with css. I believe the css id of the table should be #adopttable and from there you should be able to remove table borders, control spacing, etc.