I started bicycling to work earlier this year, and it's much less stressful than driving. My commute is about 7 miles when riding my bike, or about 10 miles when driving. Driving takes about 25 minutes on the way to work, and about 35 on the way home.
Cycling is about 40 minutes both ways. I take it a little easier on the way to work because I don't want to sweat too much, and then change in to my work clothes and do a hobo shower in the single stall bathroom. On the way home I can choose my pace, unlike when I'm driving. I can mash the pedals and let off some steam when I need to, or I can mosey home when I'm getting burned out. Stopping along the way is much easier. I'll grab coffee, or a snack, or check out a used book store I've never seen before. Plus, you are much more a part of the world when you are cycling, and you can look around more and notice things you wouldn't when driving. I'm sure a lot of these things hold true for motorcycling as well.
I'm not much of a morning person, but I find it much easier to concentrate when I arrive at work after cycling in. I also find myself much more energized when I get home, even if I'm pretty tired when I leave work. I find taking public transportation provides some of the meditative aspects, since I can reflect on my day more when I'm not driving in Chicago traffic, but bicycling allows me to combine my commute with exercise, which is amazing from both a mental health and physical health perspective.
That being said, riding a bicycle is certainly more dangerous than driving or taking public transportation. While I can take steps to mitigate the risk, I need to cross a number of bridges and go under a number of viaducts on my way to work, and the risk is substantially higher while on a bike than in a car, bus or train. I don't think I'd be comfortable riding my bike on my current commute if I had a kid.