You can hard code the links to dogs.html and cats.html into the nav and then create those pages in the root of the site, and on them use a
for loop where you loop over all the posts and show them if they match the category. Google
jekyll show posts from category and you will find a bunch of info to do a for loop.
If all you have done so far is play with the default theme, you are just scratching the surface.
Also, if you are using the default jekyll theme, the theme files are sort of hidden, which makes editing them a little difficult. I don’t use jekyll themes so I won’t be of much help, but there is a fair amount of info on how to edit them.
If you need more help post more info about your setup - ie which theme, GH repo link if there is one and a specific question and I’m sure you will get more help.