The only real danger when dyeing comes when using mordants, other than acid (like vinegar or citric acid) or alum. When you start working with mordants such as chrome or copper, you need special equipment.
Unless your pets get up on the counter or stove, you should have no problems.
RIT dye is actually two dyes in one ... one for your protein fibers and one for your cellulose fibers ... none for the acrylics! You do a lot of rinsing when using RIT dye as the dye not for the fiber you are using needs to be rinsed out.
Easter egg dyes work real well also ... they do need vinegar.
Just be sure when you are dyeing that you use hot water, but not boiling (or even simmering when using fiber that felts easily).