Nara park is really cool.
Situated in the middle of Nara, it's easily accessible and most likely, the reason you came here.
The deer here are a mixed bag. You can pat them most of the time, with no trouble. Sometimes they may bite you or head-butt you. But they are wild animals, so caution is advised.
Don't buy the deer biscuits, a group will rush you and probably headbutt you until you feed them.
A much funnier activity is to watch OTHER people buy the biscuits and watch THEM get rushed. Never gets old, hahaha.
But seriously, the park is beautiful and full of wonderful things to see. The food is expensive and sometimes absolutely horrible. Be careful what you eat.