Poodle, Maltese, and Havanese are all low-shed breeds. Other than a large standard poodle, they run small. You can find combinations/mixed breeds that include one of those. Buying a designer new breed (like a Labradoodle, a Cockapoo or a Maltipoo) can run into big money, just as the usual purebreds are. A friend often posts info on a Poodle Rescue group, so you can look into things like that instead of just your neighborhood shelter. We rescued a terrier mix 7 years ago - she was written up in the paper as a Cairn Terrier mix, but the shelter called her an Affenpinscher mix. I can see both in her. Her body shape and coloring are like an unusual terrier (Dandie Dinmont), and the local pet store owner/groomer felt the dog might have been the result of an intentional crossbreeding that didn' turn out a show-worthy dog, so she got farmed out and passed around to a few owners before we got her. She doesn't shed much, as our prior Cairn didn't. No no-shed, but certainly not a dog that has to be sold with a vacuum cleaner attached.
You have little kids, so I would not advise a dog like ours - if they've been rehomed many times, they can have emotional issues. Ours has PTSD and could not be placed in a home with kids. So go to a really good shelter or rescue group that knows the dogs and their personalities. I would shy away from dogs brought from far away as they can be stressed - but if you go to a good humane society that really looks for a good match, that's great. Our HS made us fill out a 7 page application, asking questions about family make-up, whether someone is home all day, making sure we'd planned on where the dog could go and what parts of the house she might be banned from, asking how much we thought it cost per year for a dog, what we would do if we moved to a place that didn't take dogs, what we would do on vacations, etc. It helped them assess our expertise as well as our lifestyle to be sure it was a good match. So if you are presented with that, consider it a good thing!
I would encourage you to engage a trainer unless you are highly experienced. It's good for the kids to learn from a third party "teacher" that there is a right way and a wrong way to treat a dog. Even loving dogs should never be grabbed around the neck by an aggressive hugger, no matter how well the dog tolerates it the first few times. And a crate for calm time is vital in a house with kids. And so on!
Good luck - hope you find the perfect pet!!