About washing clothes. Before modern dyes most colors would not have been color fast. Yet clothing had to be cleaned at least occasionally. I realize that layering garments would have permitted washing like colors together and that sashes, lace, and other ornamentation could be removed and resewn after cleaning. Paintings of medieval peasants show colorful but monochromatic clothing. But what of, for example, the beautiful designs of Japanese kimonos or the embroidered (?) dalmatics of Theodora's attendants in the famous Ravenna mosaic? In the absence of dry cleaning how were these items washed?
To cover the basics first, dyes before modern dyes were color fast and dry cleaning did exist. One of the most popular dyes still today is indigo. Interestingly it doesn't dissolve in water nor adhere to fabric, so how does this work? You need an alkaline solution such as lye, ammonia, or urine in order to dissolve the indigo. Once set into the fabric, water won't wash it out. Other dyes that are water soluble will still be color fast because of the process of using a mordant. The New Pocket Cyclopaedia of 1813 lists common mordants as "sulphate of alumine, oxide of tin, oxide of iron in combination with acids, oxide of arsenic, tan, &c." It goes on to say that the most permanent dyes are cochineal and gum-lac (scarlets), indigo and woad (blue), dyers weed (yellow), and madder (coarse reds, purples, blacks). Mordant can be added before, during, or after the dying process depending on the chemistry needed. It does change the final dye color. Iron oxide mordants are notorious for their deterioration of textiles, and are the reason so many black garments survive in terrible shape if at all. Basically mordants help the dye to bond to the textile and keep it much more permanent. Silks and wools do incredibly well with this, linen less so and it's harder to get a dark color set into it for that reason.
When it comes to cleaning you are very correct about not having to wash exterior pieces often (if at all). Undergarments were white for this reason, being able to be bleached and boiled to make them clean. Outer garments can sometimes be washed and submerged in water, but more often are spot cleaned based on the exterior dirt/stain. Dry cleaning by definition is simply cleaning with something other than water. For example, if you got a grease stain on silk (due to the carriage or dinner), it gets sprinkled with fullers earth, covered in paper, and a light amount of heat applied. This will draw out the stain (in modern day use baby powder to much the same effect). To remove any remaining rings you can wash the area with soap and water, washing it out with gin, and then washing it out with water. If the entire garment must be wetted to keep from getting water rings you'll wrap it up in a towel immediately to dry. Cook books often have recipes in back for laundering. And if you don't want to deal with it professional laundering can be had as well.