Here's a secret to banishing wrinkles from your laundry: Don't wash or dry so many articles of clothing per load.
If you pack so many clothes into the washer or dryer that those clothes spin as a large, cohesive ball rather than tumbling as individual articles of clothing, you're at — or over — the load capacity of that machine. (Sound familiar?) Split that load into two and you'll not only avoid wrinkles, you'll also have cleaner clothes.
Clothes need a little breathing room to move around in the washer, and without that space, each and every garment may not get the agitation and rinsing that it should. Overloading also contributes to the likelihood that your clothes may become damaged during the cycle as they become twisted and wadded up.
The same rule of space goes for the dryer. Shake each garment before you add it to the dryer, not only to untangle clothing (and release hidden socks) but to help reduce wrinkles. The added bonus of keeping your dryer loads small: Not only will your clothes dry more evenly, but they'll also dry more efficiently.