Hand washing should be done for a couple of reasons: you're washing delicate items that might get damaged in a machine, or you're only washing a couple of things and it would be a waste to run the washer. But, if you have a newer, efficient machine (like a front-loader), it doesn't cost much to run it (the electricity), it won't use much water, and it's a lot more convenient.
The process of washing cloth is to force water and detergent through it to remove soils. This is done in a machine by side-to-side agitation (top loader) or by tumbling (front loader). By hand, I have found that it works well to put clothing into a sink or tub full of water and detergent one piece at a time, and then swish it back and forth under the water to force the water through the fabric. You can also squeeze the item under the water a few times to force water through the fabric.
To rinse, you can do the same thing in a sink/tub of clear water and/or under a running stream of water. Squeeze out as much as you can prior to drying, or run the items through just the spin cycle of your washing machine to remove as much excess water as you can.
With some delicate items, as well as certain fabrics like wool, it's also best to let them air dry instead of using a tumble dryer. Wool especially will shrink a lot if tumble dried.