i too live in an apartment, and this is what i do:
leaves, stems, roots, and small branches, cut them up, let them dry out, grind them up or chop them up, and use them in the soil the next grow. (i mix it with soil, and place it in the bottom of the pots, its like slow release nutes). fan leaves also make great mulch during the summer, to keep the dirt cool. just pile it on top, a 1 inch layer over the dirt.
Sticks, branches, stalks: i do 1 of 2 things. if they are long, ill save them, and use them to support branches in later grows. shit is pretty strong. otherwise, ill cut them into 10inch pieces and give them to my dog, she loves chewing on them
i throw nothing organic out. fuck that. thats free nutes. kitchen shears, your best friend.
when it comes to actual trash, plastics, whatever, then yea, get them, put them in a bag, with nothing else (like bills for example) and drive to a shopping center, and toss it there. either way, its not illegal to have nutrients.