As for getting a mutt, they tend to live longer and have less of the genetic problems associated with most purebred dogs.
That's not actually true. People throw "hybrid vigor" around as a reason to adopt mixed breeds, but take a Golden retriever with bad hips and breed it to a German shepherd with bad hips and you're going to have a mutt with bad hips. And people who allow their dogs to breed indiscriminately tend not to carefully screen them for genetic problems in the first place (or they'd neuter them)
Nothing wrong with mixed breeds at all, and shelters and rescues are good places to look for a dog and save a life. Rescues do have the advantage of fostering them in real homes (some dogs are basket cases in a kennel situation but just fine in a home -- you'd be surprised how many dogs' behavior makes them seem like they're going to need a tremendous amount of rebuilding, but get them outside the shelter into a "real life" environment and they're lovely pets. (Can't count on it, but that's why rescues are great -- they do that part of it for you)
Most rescue dogs in my breed (boxers) are adolescent males -- they just got too big (yes, folks, puppies grow up, and if you don't teach them manners, they won't have any). Generally speaking, a simple obedience course and somebody teaching them "house manners" results in some pretty good pets, because there was never anything wrong with them in the first place -- they just never had any guidelines to live by and their owners got overwhelmed or bored by them.
You could also volunteer to be a foster family -- you might fall in love with the dog you foster and be able to adopt it. There are all breed and mixed breed rescues.
Another source of a dog is a slightly older show dog. Often breeders keep 2-3 pups from a litter to grow them out, see how they develop, and retain the best one to go forward. So there are often pups from 8 mos to 15 mos or so that showed some promise, but didn't really develop the way the breeder had hoped. As far as a pet home is concerned, usually the difference is miniscule and not noticeable (a little too gangly, too much (or too little) angulation of the legs, etc.) Often it's a cosmetic thing that only real breed aficionados would pick up on (but which shouldn't be bred because there are already too many _____s with long backs/short necks/narrow muzzles/whatever the shortcoming is. There are plenty of dogs in the world who need homes, so only the best should be bred and breeders try to select the ones who resemble the standard of perfection most closely.
Also, there are dogs who just plain don't like to show, and breeders often look to find homes for them where they can be king/queen of the household instead of one in a crowd. Like humans, some dogs are homebodies! (Most showdogs adore it, however, and can't WAIT to get in the van/motorhome and go where the action is!)
Likewise, the pup may not have the health test results that the breeder needs to carry forward. There are heart conditions that aren't fatal for the dog, but if bred generation after generation get worse and worse, so a dog that shows the condition on an echocardiogram shouldn't be bred because its offspring may have a more severe form that WOULD affect lifespan. Similarly, a breeder looking to improve hips in her line would elect to keep the pup with the superior hips, all other things being equal, even if the pup she elects not to keep isn't likely to ever have any hip issues.
So you can always go to a dog show and chat with people in the breed you're looking for, asking if there are any older puppies or young adults somebody's looking to place. You're likely to get a dog that's house and leash trained, socialized and ready to step into life as a companion.