Ok, so we live in NYC, where basically only rich people have laundry in their units. We rely on laundromats, and it can definitely get expensive. Here's what we do:
1. We have two laundry baskets, one is for things that must be dried in the dryer, and one is for things that don't get dried in the drier. Line drying is actually better for clothing, but of course, for things like bath towels, sheets, and comforters, this isn't an option. This is also great because my fella (who does the laundry) doesn't have to memorize what goes where. My silky tops, bras, stockings, and jeans are line dry, but my sloppy t-shirts and socks/underwear go in the dryer.
2. Make indoor laundry lines. We air dry about 75% of our laundry. The largest items get put on a drying rack in the bathtub. The smaller items go on our clothing line which is a piece of rope that we tie to the bed frame, stretches across the room, and onto the far wall. Getting the water out is definitely a pain, but it just takes time. You have to delicately squeeze the water out of each piece by hand. Never wring clothing, just use pressure to squeeze the water out. Then shake out each piece, so that it doesn't get wrinkled, and hang it. If water dripping is a huge issue, then line the floor with a tarp.
3. Timing. We do laundry on Saturday or Sunday morning. This way the wet laundry comes out by noon, and has a good 10 hours to dry, plus keeping the windows open or a fan on them helps. If it's still a bit damp, we leave it over night. By morning it is always dry.