No hard and fast rules here. And 5 months old is fine for solids, if she's showing that kind of interest! You can start fruits, veggies, or other cereals whenever you're ready- as long as you wait a few days in between (like you said). I'd wait on wheat cereal until later, but oatmeal is fine! I started giving my son two meals a day when he started getting mad at me for eating in front of him! :) I think it was actually pretty early, around 6 months. And 3 meals came naturally after that, based on his interest.
And we actually started with avacadoes, bananas, and sweet potatoes before cereal! (One at a time, of course.) My guy hated cereal unless it was mixed with another flavor, anyways.
Just follow your baby's cues, like you're doing. No need to stress, as long as you're careful about possible allergens!