We have three small kiddos (ages 5.5, 2.5, and 4 months) and these are our favorite places to hike:
Montclair Railroad Trail (Oakland, CA):
This path is mostly paved (we brought the stroller), super flat and really scenic, with several benches along the way to stop and rest at. It’s about a mile in either direction from the road just behind Montclair village (park on Lasalle Rd or in the parking lot next to the road). If you go on a Sunday you can stroll down to the Montclair Farmers Market for a post-hike snack!
More info here: http://www.montclairrrtrail.org/
Sibley Regional Park (Oakland, CA):
If you park at the main entrance where the bathrooms are located and hike up it is a slightly steep climb but super scenic! Less stroller-friendly and lots of dogs (could be a plus or minus depending on your family). Off path are beautiful hills and vegetation and so much space for the kiddos to explore. Also, my oldest thought it was pretty cool to be hiking up an old volcano, and by the bathrooms in the main parking lot there are some cool fossils and an opportunity to learn more about the history of the park!
More info here: www.ebparks.org/parks/sibley
Big Trees Trail in Joaquin Miller Park (Oakland, CA):
This trail is mostly flat and you feel like you are in an enchanted forest surrounded by the redwoods. My girls loved exploring the ferns, trees and little off trail hiding places. It is also very dog-friendly and can be hiked as a loop. Total distance is under 2 miles and we made it almost the whole way with all the kiddos. We brought the stroller but unless you have a stroller with good tires like a jogging stroller it is probably easiest not to bring it
Inspiration Point (Berkeley, CA):
The best hike for strollers, skooters, and kids on wheels. Parking can be a pain, but it is worth the struggle, because the views are amazing and the path is completely paved. The trail can be really busy on the weekends but it is a nice, easy, family trip with young ones in tow.
More info here: https://www.yelp.com/biz/inspiration-point-berkeley
Point Pinole Shoreline (Pinole, CA):
Worth the drive and a scenic fun place to spend the day. This place has it all: beach, a bridge to watch trains over, hiking trail, and playgrounds. The beach here is rocky but shallow and fun for the kids, on the San Pablo Bay and the hike starts with a small hill but is fun and can be done as loop overlooking the San Pablo Bay. My girls loved exploring all areas, including the playgrounds which are sand-based and surrounded by grass to lay in.
More info here: https://bahiker.com/eastbayhikes/pointpinole.html