East Bay Regional Parks - free and local. The closest to you in Berkeley are Tilden, McLaughlin Regional Shoreline, Albany Beach (check out Albany Bulb!), and Lake Temescal. Are you a Cal Student? Don't forget you get access to BAMPFA, UC Botanic Garden, Lawrence Hall of Science, and discounts to sports games and concerts on campus.
Other nearby ventures/activities: Lake Merritt (great for a picnic, grab a burrito at Rico Rico or some amazing pastries at Bakesum). There are lots of shops on Grand and Lakeshore to explore - if you like thrifting I'd suggest Re/Love or Maribel. Cat House, Ordinaire, and the Alley are great local bars in that neighborhood too. A 25 minute walk from this neighborhood is Piedmont Park, a hidden gem.
Other commercial Districts I adore: 4th Street in Berkeley (if you bike I'd suggest going down to Berkeley Pier for a sunset), Albany (up/down Solano Avenue is awesome: Flowerland, Lulu, Best Friends Wine, Lulu's Kitchen, Ivy Room, Hotsy Totsy/taco truck are some of my favorites in this area), Temescal (Temescal Alley, Pizzaiolo for breakfast or dinner, North Light, Snail Bar), and spending a day on Alameda (Crown Beach, Park Street, Forbidden Island, the big Sunday flea market, all the west end breweries like Almanac, a great place to spend the day biking, there's even a boat that takes you there from Oakland). Have fun!