Disposables have a use, but so the textile rags are nice too.
There are a bunch of options when it comes to cleaning the rags. The first is obviously the local laundry. Check the ones that are closest to you to see if they offer a washer that is for work clothes or rags only. If they don't, it will be up to your conscience to decide whether or not running rags through a normal washer is going to win you any points with your neighbors. I have done it in the past myself, but I made sure to presoak all my rags for a few days in a five gallon bucket of dawn dish soap and water before I brought it down to the laundry. Then after I was done with the rags, I ran my shop clothes through that same washer to help clean the machine. When the clothes were done I checked the drum for any remnants before leaving it to someone else. You could also pre-soak and wash at home depending on how much your wife appreciates it. If you have the space and the hookups you might consider running a separate used washing machine just for your shop rags and clothes.
Another option is uniform services. Most uniform places will collect rags and wash them on an exchange basis. That means if you have brand new rags and send them in, you probably won't get brand new rags back. You usually would want to hold off on submitting your rags until you have a significant amount to swap with the company so storage becomes another problem for this route. You would want to make sure you store all your dirty rags in a sealed metal container to reduce the risk of fire.
Never, ever run rags through a dryer. Even after they have been washed the best way to dry your rags is via a clothesline.