A lot of the "how" is really dependent on quantity. I use a color cup almost exclusively for modeling. Any residue I have simply goes into a couple napkins, and then into the trash can. For larger quantities, you could pour them into some sort of old container, set it outside, let it evaporate and throw the solids away as trash. Another option would be to pour your paints into a container, put a lid on it, and when full take it to your local recycling/hazmat center or throw it in the trash; or if your original paint bottle is empty, pour your residual paint back into the bottle, cap it, and throw it into the trash. Interestingly, there aren't any real EPA regulations governing small quantity users like homeowners/hobbyists, etc. Obviously, pouring solvents down the storm drain is incredibly illegal (fines + jail time), but there are a lot of options available for getting rid of your waste paint.
In terms of health concerns, I think it's important to get your paint residues outside as quickly as practicable, unless they are in containers with lids firmly in place. You don't want thinners and other solvents sitting in a trash can evaporating and filling your hobby room, family room, etc. with a lot of unnecessary vapors. Oh, yeah, and keep all your solvents, rags, etc. distant from any heat sources: spark, flame, sources of static electricity, etc. Fire is certainly a concern, but just as importantly, heat causes increased solvent volatility, and consequently higher potential inhalation exposures. Make sure you maintain adequate ventilation, also.
Hope this helps
Gip Winecoff