Let us start with the clotheline. If you don't have one, get one. If you live in a subdivision that doesn't permit them, hang one in your attic, porch or use a drying rack inside, or do a little agitating and get the rules changed, but lose the dryer. I had a friend from Indonesia who said that of all the funny things she'd learned about Americans, the idea that we used machines to dry our clothes, which the air and the sun did for free was absolutely the weirdest. She couldn't get over the idea that we were that crazy. And she has a point. The average dryer costs most families $80 per year. That's a lot of money for something free. And the greenhouse gas emissions are signficant.
Now I don't find hanging laundry to be a hardship at all. I can get a load up on the line in 4 minutes, the laundry smells better, and while I'm doing it, the kids can help, I can watch the birds in the trees - frankly, I find it to be one of the most pleasant chores I do. Now it is true that things are softer when they come out of the dryer - and some things, like towels, can end up on the crunchy side if you have hard water. A little vinegar in the rinse, and hanging on windy days helps. But the thing that helps the most is simply getting used to it - it won't take very long before you'll stop expecting everything to be soft. And everything smells better on the line, and many things come off the line nicely crisp - oxford shirts, sheets and tablecloths are much nicer off the line. If you are really having trouble with the transition, you can throw your already 90% dry clothes in the dryer for a few minutes to soften up, but it is much easier just to get used to it.
In the winter, you can freeze-dry clothes outside (they come out softer), or hang them up inside on an indoor clothesline or drying rack. I often just throw them over the shower curtain rack as well, but I'm a notorious slob, so you probably shouldn't emulate me, or should at least hide the clothes when company comes over.
The next thing to lose is the dry cleaner. Conventional dry cleaning uses incredibly toxic solvents to clean your clothes, which are neither good for you or the environment. It is also extremely energy intensive. Treehugger has a good analysis of the issues here: http://www.treehugger.com/files/2007/02/ask_treehugger.php. If you have an environmentally friendly dry cleaner near you, by all means use it. If you don't, avoid "dry clean only" clothing like the plague. If you must wear it, do so as infrequently as possible and go as long as humanly possible between dry cleanings - air and spot clean instead. But if you don't have to wear an expensive suit every day, choose clothing that doesn't need dry cleaning. Most natural fibers (preferrable in many cases to petroleum based) can be hand washed - cashmere and wool, for example.
I don't think chemical fabric softeners are a good idea for anyone - dryer sheets and chemical fabric softeners contain a host of chemicals - here's a list of some of the chemicals and their potential consequences. http://www.ourlittleplace.com/fabric.html. Just do without them. And you'll want the lowest impact detergents, with the least packaging you can get. At a minimum, buy bulk powdered laundry degtergent in cardboard, use as little as you can and still get the clothes clean, and recycle the packaging. But if you can afford it and have access to them, there are a host of non-toxic, environmentally friendly detergents out there, or if you make soap, you can make your own laundry soap and know that the ingredients are quite friendly. Traditional washing soda, available at some stores is also very low impact. Borax is considerably better than bleach on your whites, and is available at my grocery store.
The next thing you can do is reduce the sheer amount of laundry you need to wash. Towels and sheets can be washed less often. Clothing can be reworn, perhaps aired if it doesn't smell fresh, and washing avoided. Don't just toss things in the hamper - make sure they really need cleaning. Put out of season clothes away in places that get decent air circulation if possible, to avoid a need for rewashing. And spot-treat stains to avoid multiple washes.
Toilet training makes a huge reduction in parent's laundry - I'm not exactly the poster child for getting your kid trained early - my oldest disabled child is not yet trained, my second was nearly four, and #3 just trained at 3 1/2. But the sooner you do this, the sooner the cloth diapers go away, and that's quite an incentive. Obviously, you can't make a kid toilet train, but if you can do it, it is worth devoting some time to.
I have done all my laundry in cold water since I was a kid, and my clothes come out fine. I find it requires a little more attention to stains since my water here is very, very hard, but there is really no need to do laundry in hot water. We wash diapers in cold as well, and have had no problems with rashes or diseases. Hanging them in the sun makes an excellent natural antibacterial.
And then, there's hand and non-electric washing. It doesn't have to be an all or nothing thing - you can, for example, hand wash a sink full of underwear and socks in a couple of minutes, and thus perhaps delay your need for a machine load. Or you might consider hand washing everything. For one person, hand washing everything but towels, sheets and jeans would be entirely feasible - I know, I did it for years, simply because I couldn't afford the quarters for the machines. I reserved the trip to the washer to once a month. And when I finally had a washing machine, I was struck by how much more often I was using it.
There are a whole host of ways to do laundry without electricity. Lehmans sells a very small pressure washer, great for a small household or a few diapers. It costs less than $70. Or you could spend a ton and get the James Handwasher, wringer and rinse tubs. Or you could use a free plastic bucket from the grocery store and a plunger, or simply soak things a long time and rub them together with your hands. Or you could do what Colin Beavan and his family do - grape stomp your laundry in your tub. http://noimpactman.typepad.com/blog/2007/04/a_grape_stomper.html
My personal fave is the bucket and plunger method, but honestly, I went years using a bathroom sink, some water and a little detergent, and life went along just fine. I admit, I like my washing machine, but it is good to know I can get along without it.