Should is a difficult word.
I think you should do what you want, not what masses or a stranger says.
You say nothing about yourselves in the post or your profile so I can only guess if you prefer buses (you can do a bus depending on where you are) or riding bicycles (you can do that too, but I don't know if you are fit, or elderly, or have 10 children with you).
To see the Appian Way the best path is the Appian Way. I don't understand your question number three.
Rick Steves in his Rome guidebook spends many pages on the Appian way and how to get there and what to do.