Sleeping Giant is beautiful state park only 20 minutes outside of downtown New Haven. There are a ton of trails, designated picnic areas at the base, a castle at the top, a hidden cave, climbing routes, and so much more. Red trails go North and South. Colored trails go east and west. Blue (my personal recommendation) is the hardest and goes straight up the face. Total elevation 739ft (225m).
The main entrance is directly across the street from Quinnipiac University. There is onsite paid parking, but the lot closes at sunset. The park service can be strict about ticketing cars still parked in the lot after dark, so pay attention to the time. You've been warned. Some trails can be difficult, so be prepared and bring some water and a snack as well as durable hiking clothes. A camera is always a plus.