Comparing the families Canidae and Felidae, which have the shorter evolutionary distance to the order Rodentia? Less sophisticatedly put, are cats or dogs more related to rodents?
Here is exactly the kind of question where tolweb.org is your best friend. tolweb.org is one of the most updated phylogenetic tree online. It also contains tons of reference, very useful to understand controversy over some nodes. onezoom.org is another online source that offer you to seek through the tree of life. Also, onezoom shows only the tetrapods (incl. mammals, birds, amphibians, lizards, snakes, turtles). However, onezoom has an awesome display and is a great tool for displaying diversity (they use the total length (in years) of all branches within a clade) within a clade and the threaten index (IUCN (red list) categories).
Dogs, Cats and Rodents
You will find the placental mammals here (tolweb). As you can see, the cats and dogs are both carnivora, while a rodent is a Rodentia (as you said). This means that cats and dogs have a more recent common ancestor than does either the cat or the dog with any rodent. Stated differently again, cats are not closer or further to rodents than dogs are. There is exactly the same distance between cats and rodents than between dogs and rodents.
One onezoom, the carnivorans is presented as a sister group to the ungulates. I have no idea what representation is the most correct but in any case, cats and dogs are carnivorans and share a common ancestor that is more recent than the common ancestor shared by carnivorans and rodents. So again, there is exactly the same distance between cats and rodents than between dogs and rodents.