Welcom to the forum and congratulations to your new family addition.
With mixed breeds it is sometimes difficult because they've got a bigger genetical variety in looks and character than a properly bred dog.
Your advantage towards a mixed bred dog from a rescue is however, that you know both breeds in your little mutt.
this is great because you can explain why your dog is doing this or that and can brace yourself for the "worst-case" insteasd of just being surprised with it.
So let's start with the breeds
the Poodle is a hunting dog that was bred for hunting water birds. Like the retriever for this Job it has to work together closely with their human. this means the most poodles have certain amount of WTP, which is great.
but the most of them have also a bit of hunting drive, which (depending on what you want) can be good and bad. Since the Poodle is orientating closely to the owner, it is often manageble to the point it can run free when trained though.
Different than retrievers the Poodles seem to be a bit more aloof and less interested in meeting strangers.
Eventhough the Poodle has WTP, it is pretty intelligent and when bored, can become a real handful.
he learns fast and is relatively easy to train, but because he's an easy learner he's also pretty fast at learning things you don't want them to learn.
the "fur" is more like hairs and will grow without falling out, which makes it a great dog for a allergic people...but even a poodle produces saliva and it'll clean his fur by licking, so the saliva is in the fur. A lot of Allergics have a contact allergy to this saliva, so these persons should still wash their hands after petting the dog, or let them lick their hands.
the fur also needs a bit of care, like trimming, and combing.
Now the Schnauzer.
until the beginning of the 20th century the Schnauzer was just one coat-type of the Pinscher family. that's why the old name of the Schnauzer is Rauhaarpinscher (wirehaired pincher).
this means the closest members of your schnauzers familytree are the Dobermann, the german pincher and the small pincher. so generally, you could see the him as a German Pincher with a beard.
...well not really.
The job description of the Pinscher family was pretty variable.
One was guarding humans carriages and properties, similiar to the Dalmatian and the most farm dog breeds (Appenzeller or Hovawart).
For this job he had to be a bit distrusting towards strangers and bind himself closely to his property and his master.
Most Schnauzers (giant schnauzers) that i know, are strict one-person-dogs, similiar to the Dobermann (but a bit less sensitive to stress).
The small Schnauzer was used for hunting small mammals, like muskrats, rats, mouses and beavers, so his work was similiar to a rat terrier, the Prager Rattler or the spanish Ratonero breeds.
for hunting these animals, he had to be intelligent, independent thinking, strong-willed and resilient. he also had to have a certain drive to kill these animals.
So your worst-case scenario is a dog, that has a strong hunting drive (both breeds are hunting dogs), a strong territorial and guarding and protection drive (Schnauzer), that is stubborn and independent thinking like a Schnauzer and and hyper and sensitive like a poodle.
it also doesn't like people, except this one person he adores, and in ideal condition at least grudgingly except a family around that person.
the ideal case scenario would be a easily trainable, robust companion dog, that likes to work with you and has a medium hunting drive.