I've noticed something interesting, in an it-kind-of-irritates-me way. "Dry clean only" on the care tags of clothing that has no reason to need dry cleaning.
I have a coat that claims to be "dry clean only". It's made of polyester, which is impossible to damage with water or laundry detergent. The weave of the outer shell makes it require some minimal care in washing, rather than throwing it in the machine, but there's absolutely no reason not to wash it by hand, which I have done several times.
Dry cleaning is expensive, and everything that gets dry cleaned smells like a chemical swamp. Nasty! Why are so many clothes unnecessarily labeled for dry cleaning? I can't say I know, but the best I can come up with is that they're trying to make it seem fancier than it actually is, since dry cleaning is usually for really nice clothes.
I'll cut right to it. There's only one material that always, always should be dry cleaned rather than hand washed, and that's leather. If it's not leather, unless you really want to make 100% certain that it won't fade, don't bother. Even leather can usually be safely spot cleaned with a soft damp cloth, though it's always a good idea to test it in an inconspicuous area to make sure, and never ever stretch dampened leather unless you're trying to shape it!
Common materials to see labeled as "dry clean only" are silk and cashmere. While some of the particularly fine silks can be damaged by water, 99% of silk clothes are just fine being hand washed. If it's not high-end designer, chances are good it'll be just fine. Test it if you're worried by wetting a hidden spot, like an inner seam or facing panel. Cashmere will never be damaged by water, though like wool it will become felted and shrink in hot water, so wash it in cool. Both silk and cashmere are delicate materials and should be treated gently, never put in a washing machine, and always allowed to air dry. Don't wring them out!
The best way to wash delicate fabrics is to dissolve the detergent in water, swish the garment around for a minute, just enough to get it thoroughly wet, then let it sit for five minutes. Swish it again, drain the soapy water and fill up the sink with fresh water, repeat the swish-sit-swish setup. I like to rinse it twice to make sure of getting all the detergent residue out, as it can be a skin irritant.
If the garment is wool, cotton, or any synthetic material, unless it's particularly delicate or has beading/rhinestones on it, it's fine to machine wash (though I like to hand wash most wool to avoid pilling). Pay attention to the 'cold' or 'warm' directions on the care tags; if it says to wash it cold than it will almost always shrink if washed warm, particularly cottons or wools. Of course, this can work to your advantage when there's a great shirt that you can only find in a size a bit too large!
Cottons and silks are the fabrics most inclined to wrinkle. Unless they're a blend, they'll probably need ironing. Use the cool setting for silks, cottons can handle higher heat. It's a good idea to iron anything inside out, to avoid making it weirdly shiny.
Hopefully this is helpful!