I grew up with toy poodles. My mother has one now. We've always done our own grooming.
I'm sure there are some wonderful products out there that would make it a lot easier, but what we do:
Brush her out good before bathing. The bath will just make the matts worse.
Bath her good with a mild soap. We use baby shampoo. Then, as someone said before, a good conditioner....let it sit a few.
Rinse well.
We brush as we dry, a section at a time.
trim the feet and base of the tail with clippers. Use longer blade on the face or just scissors (whatever is more comfortable for you)
The little flea comb is good for the tears.
The body, we always use scissors on as needed. Shorter in the summer, letting it grow in the winter.
The biggest difference I see between my yorkie and Mom's poodle is the brush.
The slicker brush works better for the poodle hair. It's curlier and thicker than my yorkie's.
This is the type brush I'm talking about for the poodle.
We do her nails last and then spritz her with a little doggie perfume. She loves her grooming time and will absolutely ROLL in the yard to get dirty if she hasn't had a bath and wants one.
Good luck!