I've just been through this process :)
Please no "doodles", pugs, or anything with a name like morkie, shi poo, jug, or a cross being passed off as a breed.
They are not hypoallergenic, and the fashion for these dogs is driving puppy farms and irresponsible breeders.
If you want the non shedding, intelligent qualities of a poodle, buy a poodle.
Rescue wise, look at sighthounds- whippets, greyhounds etc, or staffies. There are quite a few staffie pups in rescue.
Staffies are actually fantastic dogs. I seriously considered it but they are just to big and bouncy for my small children. But do please look, there are so many need homes.
We ruled out sighthounds as we have small furries and like to go for 2-3 hour walks rather than short fast runs.
Other types of dogs are actually very hard to get hold of via a rescue. I was filling in 3 application forms a week for nearly a year, country wide, and got nowhere.
Manytears often have rescue puppies from puppy farms.
In the end i went to a kennel club assured breeder- and to my point above a kc reg puppy is often lower price than a chipoo or schnoodle. I went for a yorkshire terrier, a breed i am familiar with, i know it can go for long walks, or is happy with a turn in the garden, that they are intelligent and trainable, and that the coat is non shedding, but needs care.