Look for double duty pieces--ottoman that has storage inside, Under cabinet appliances, decorative baskets with lids for toys or magazines to keep public areas neat. I don't know if this is space saving or not, but I swear by wall lamps. They keep nightstands & table tops free & therefore not liable to be knocked over & broken by children or cats.
Can you use the upper reaches? I have a lot of decorative tins collected over the years? Dh made a long shelf a few inches below the ceiling, right above the curtain rod in one room & they're all up there now, cheerful to look at but not in the way. In the same room I used antique aprons collected from thrift shops with clip-on rings as curtains.
Oh, I almost forgot. We rent, so I have a number of inexpensive devices that go over the tops of doors with no hardware involved to store things on. One has 12 hooks & is over the back door for jackets & hats, another on the pantry holds aprons I use, in second bedroom there is one with off season clothes on it. You can also buy a variety of coated wire baskets you hang on a pantry or cupboard shelf to increase your space available. Those are great.