With their short-haired, smooth coats, Smooth Fox Terrier grooming doesn't take much work at all. These dogs' coats don't shed much, and they're in reality too short to mat or tangle, so they'll only need brushing once every week or two. Regular baths aren't necessary--only if your SFT gets especially dirty or stinky. And haircuts aren't required at all.
Using a regular bristle brush is fine for brushing these dogs--just make sure it's not the same one you use for cleaning, as the chemicals in cleaning products will irritate the dog's skin. (It's best to have a bristle brush specifically for brushing your SFT.) To brush a Smooth Fox Terrier: first wet the coat with mist from a spray bottle, then go over the coat section by section, moving in the direction of hair growth. And that's about it!
If you're following the brushing with a bath (and the SFT will need a thorough brushing prior to bathing): you can use either a bathtub or an outdoor kiddie pool and a garden hose. First wet the coat thoroughly, then apply a quarter-sized amount of canine shampoo (not the kind for humans, which will dry out the SFT's skin) to the dog's back. Lather well, working downward as you go--and don't forget the legs! Finish with the chest, neck, and head, making sure to avoid getting soap in the dog's eyes. Rinse completely, towel-dry, then give the coat another quick brush-through to make it look neat and clean.