Woollens are one example. Many can be washed at home, but certain cautions must be taken or you'll end up with doll clothes. First, cold water must be used. Even warm water could cause shrinkage. Most delicate fabrics have tags recommending no scrubbing or wringing action should be used when hand-washing. That's why I have a clean toilet plunger I use on laundry only. The plunger gently forces water through the fabric with a gentle cupping action. I'm careful not to plunge completely, though, as the greater pressure can distort fabrics.
Another thing to remember when hand-washing is to rinse thoroughly. Cold water makes it more difficult to get rid of suds, so a couple or few rinses may be required. This is important. If soapy residue remains, fabrics will stiffen and be uncomfortable or irritating on the skin. To help the rinsing process, add a generous amount of white vinegar to the final rinse water. It'll not only help dissolve any remaining soap, it'll soften the clothing, prevent lint from clinging, and neutralize mildew and/or other odours. If you have any sensitivity to wool, you can also add a capful of creme hair rinse to the final rinse water and they will be softer to the touch. Glycerin can be used in place of creme hair rinse, but more will be needed to achieve the same result.
Before washing, test the fabric along the inside seam to ensure the colours won't bleed. If you choose to hand-wash non-colourfast items, make sure to wash them individually to prevent cross-colouring. It's possible to stabilize the colours and prevent fading by adding Epsom salts (1 teaspoon per gallon) to the final rinse.
Drying hand-washed clothes can be challenging, particularly as most cannot be wrung out. Again, I use my handy laundry-plunger to squeeze as much water from the fabric as possible. For clothes that could stretch, you'll want to lay them flat to dry. A blanket or towel works well to absorb the moisture, but the drying item should be kept in a well-ventilated area so the fabric dries as quickly as possible. A fan helps speed the process. A suspended drying rack will greatly improve the drying time by allowing the free flow of air around the garment. For clothes with resilience (non-stretching) hanging them will achieve faster drying but, again, hang in a well-ventilated area and make sure you have a drip catcher in place.
One drawback of hand-washing is the need for ironing. Woollens, of course, won't require ironing, but most other hand-washed garments will. Iron delicate fabrics on a low setting to smooth out any wrinkles. If the material is synthetic, use a piece of brown paper (cut from a grocery bag) between the iron and fabric to prevent shine developing. Ironing with brown paper will also enable a slightly higher temperature if a crease is desired (e.g. dress slacks.)
Although there is money to be saved by hand-washing, the main reason I duck the dry-cleaning is smell: I just can't stand the chemical odours that accost the senses upon entering the shop, and waft off the garments I bring home. So, for me, the greatest bonus of washing clothes by hand is the air-freshening attribute: damp clothes, smelling of mild soap, actually deodorize the room they're hung in. They smell great when worn, too!
Ultimately, the cost of dry-cleaning influences how I shop for clothes. I consider the garment's price to be not only what's printed on the sticker, but its lifetime cost of cleaning, too. Many items of clothing, unable to withstand that cost-analysis, never make it into my closet.