For your list of what's happening often free, from live music with renown artists, to movies projected in Brooklyn's new parks, out under the starts, check these links.
Prospect Park Bandshell: http://bricartsmedia.org/performing-arts/celebrate-brooklyn/celebrate-brooklyn-prospect-park-bandshell
Brooklyn Bridge Park: http://bricartsmedia.org/performing-arts/celebrate-brooklyn/celebrate-brooklyn-dance-parties-at-brooklyn-bridge-park
Summerstage (listings for all 5 borough city parks): http://summerstage.donyc.com/
I plan to take in Cibo Matto, whom I've never scene perform together at Celebrate Brooklyn in Prospect Park and I've got more in store.