Because these hybrids have two parent breeds that can have quite different coats, the amount of Jack-A-Poo grooming needed will depend on the coat type of the individual dog. If a Jack-A-Poo's coat is more like that of a Poodle, it won't shed much, but will easily mat and tangle and will likely require frequent trimming; these Jack-A-Poos will need brushing 3-4 times per week, if not daily, and baths about once a month. It might be a good idea for owners of these dogs to use a professional groomer for haircuts and other grooming needs.
For Jack-A-Poos with "Jack Russell"-esque coats, brushing will be needed only once per week, and bathing only when necessary. Haircuts won't be needed at all, but the coat may need to be hand-stripped two or three times a year; again, you may want to rely on a professional groomer for this time-consuming task.
Most likely, though, your Jack-A-Poo is somewhere in between the above-described coat needs, so in that case it'll need twice-weekly brushing, baths every 6-8 weeks, and optional haircuts every 2-3 months.
Brushing a Jack-A-Poo's coat, regardless of its type, will require both a slicker brush and a 2-in-1 comb. To brush your Jack-A-Poo: first wet the coat with water mist from a spray bottle, then starting at the shoulders, go over the coat section by section with the brush. If you encounter a tangle or mat, first try working it out with your fingers, then continue working through it with the comb.
If you plan to follow the brushing with a bath (and your Jack-A-Poo will need thorough brushing prior to bathing): if small enough, the dog can be bathed in the kitchen sink. If not, you can use either a bathtub or an outdoor kiddie pool and a garden hose. Be sure to also use canine shampoo, as the kind made for humans will irritate the dog's skin. First wet the coat thoroughly, then apply a quarter-sized amount of shampoo to the dog's back; lather thoroughly, working downward and outward as you go (and don't forget the inside of the legs!). Finish with the chest and neck, then use a washcloth to clean the Jack-A-Poo's head, ears, and face. Rinse completely, towel- or blow-dry, then give the coat another quick brush-through to make it look clean and neat.
Haircuts may be necessary if your Jack-A-Poo's coat favors the Poodle's. With some practice, you can learn to do it yourself--but it's a good idea to take your Jack-A-Poo to a professional groomer at least once, regardless of the dog's coat type. The groomer can demonstrate the various grooming techniques needed for your particular Jack-A-Poo, and can provide coat maintenance tips in general.