It is relatively safe to explore the city, so go out and experience it. The only part which you have to be more careful is the northern part near the railway station.
The Chunxi Road area in Chengdu is famous for shopping, especially for branded and modern goods. A lot of people, especially the yound and trendy, hang out in this area during the weekends.
Check out the People's Square. I was fortunate to witness the peaceful gathering of soccer fans there minutes after China qualified for the first time in history to play in World Cup 2002.