I know it sounds tedious, but you will get a lot more life out of your bras if you hand wash them. A washer and dryer are very harsh on your delicates and you will end up having to replace them more often. Here are some things you can use to help out.
No Rinse Delicate Wash
I swear by this product. I use it for all my wool hand-knits and it does a great job on my delicates. Unscented, but there's also lavender.
Yes, a salad spinner. Think about its purpose. Removing excess moisture. It's exactly what you need and it's just the right size.
Bras should never go in the dryer. The heat can distort the foam and fabric, which can alter its shape and will definitely reduce your bras life. A drying rack costs around $20, which is way cheaper than a new bra. A folding one is fine, but if you're concerned about keeping your bra's shape, try a flat sweater dryer.
If you do wash your bras in the washer (but of course, never using the dryer), a lingerie bag is a must. It prevents tangles and snags, and makes it easy to grab all your bras out of the wash at once. So cheap, it's a no-brainer.