Wool is great for biking. That's what I wear most of the time, in various weights for various temperatures.

You need to get clothing that does handle washing in water "washable". If it says "Dry Clean" there is a good chance that water washing will leave you with a lot of long lasting wrinkly stuff. I believe it's at least partly a function of how the wool is spun into yarn, or perhaps how the yarn is knitted or woven into cloth.

I do washable wool in the machine, mine actually has a 'wool' cycle so I use that, with Woolite. I usually hang dry after washing. Socks seem to get into the regular load and into the dryer a little more often and hasn't really been a problem with them.