Overall, Miniature Australian Shepherd grooming doesn't take a lot of work. The medium-length, thick double coats do shed some (especially during the spring and fall), but the coats aren't known to tangle and/or mat much, so Mini Aussie grooming is actually pretty easy. These dogs will need brushing once or twice per week, they'll need baths every 2-3 months, and depending on your preferred hairstyle, they'll need clipping/scissoring about every three months.
To brush a Mini Aussie: first wet the coat with mist from a spray bottle, then use a stiff bristle brush (or a pin brush) to go through the coat, starting at the shoulders and working backward and down. Be sure to brush in the direction of hair growth; finish with the chest, underbelly, and legs. If you plan to bathe your Mini Aussie, first give the dog a thorough brushing, then use canine shampoo (and conditioner, if desired; both are available at pet stores) and lather the coat starting on the back and working down. Rinse well, towel the dog dry, then give it another quick brush-through.
If/when your Mini Aussie needs a haircut, you can take it to a professional groomer; depending on your dog's hairstyle, you can also learn to give the cut at home, either from instructional videos on YouTube or other sites, or from the groomer themselves.