L.M.
My kids aren't yet, but a word to the wise.... Don't flush the moist wipes, even if they say "flushable." My mom had serious plumbing issues in her new house because my grandmother (who lives with them) uses them and they just don't disintegrate like toilet tissue. The plumber told her that none of the "flushable wipes" are endorsed by plumbing companies for a reason.
For my son (he's 4, mostly potty-trained and I use regular baby wipes on him after a BM), I put the used wipes in the diaper genie I am still using for my daughters (who are still in diapers.) When the girls are potty trained, I will keep the diaper genie in the bathroom. Perfect receptacle for "that time of the month" ya know?