I was going to propose the East Bay Regional Parks as offering short trips including some around lakes but when I checked their website,
dogs are allowed only during the day time. I suspect it may be, in part, because of their 20-30-person group camps. I know I hate being a campground when someone else's dog is barking all night.
Pity for you, as you could get to many of those parks on public transit - Lake Chabot for instance, from Bayfair BART station and then walk or talk AC Transit over Fairmont Drive. The campgrounds are on the other side of the lake, about 4 to 7 miles from the road, depending on which way you walk.
So store that info away for some time you are looking for a day hike with your dog. Or backpacking without your dog. The shortest loop of Lake Chabot is about 9 miles. If you go through Chabot, Redwood, Huckleberry, Roundtop, Sibley, Tilden, Wildcat Canyon and return, that's more than 100km. It took 21 hours when I last did, but I was half my age then.
While a little further away, the Ohlone Wilderness trail is more remote and a much longer road-free trail.
But again, dogs only during daylight hours.
Get out early, hike hard, sleep in your own bed.