Our city has a lot of diverse cultures. There is usually a festival or celebration of some sort somewhere. Not all of them are religious. There are a lot of strawberry teas and strawberry socials going on right now, particularly at some of the smaller living museums and preserved historic homes. Those smaller venues offer similar events throughout the season.
There is a public park that offers a series of free summer music concerts (mostly classical, but some jazz and some world music) on one weeknight (I think it's Thursdays) and Sunday afternoons.
Shakespeare in the Park has started. We like to take a picnic dinner, picnic blanket and some air mattresses for the evening. Every year we attend with the same friends.
It can be fun to attend some of the local sporting events like a fun run or a soccer or cricket game or baseball tournament as a supporter. Our city has professional baseball, football etc. but the semi-pro and recreational games can be a lot of fun to watch.
For awhile we lived in a city with a wonderful Children's Festival. It was held over 5 days. There were live music and theatre performances, storytelling tents, craft areas, face painting and so on. It was an awesome event.