"i wash my hands before the meal" INCORRECT. "i wash my hands before tea" CORRECT
"Meal" can mean either breakfast (el desayuno), lunch (el almuerzo), or dinner (la cena).
"Tea" is the English spelling. "Té" is the Spanish spelling. They both mean the same thing.
Technically I think your sentence is 'before eating dinner' but I agree that the overall meaning is the same.
Grammar isn't really my strong point but I believe the only time you don't need a preposition after 'antes' and 'despues' is when they effect the entire sentence (as adverbs I think). Por ejemplo:
Lo hice antes. = I did it before
Usually 'antes' and 'despues' will take prepositions after them (like 'de'). I think this happens when they are connecting two nouns like in Duo's sentence above (hands and dinner).
I understand the "me lavo" structure but can it also be: "Yo lavo mis manos antes de la cena."?
If you leave off the reflexive pronoun, iPrash, you can use "lavar" when you are speaking of washing anything BUT people. When you want to speak about people washing themselves, you must use a Spanish reflexive pronoun.