Botanical Garden
The UC Berkeley Botanical Garden, located in the hills above campus, is a great place to go for a walk to check out cool plants from around the world. Entry is free with a Cal student ID. If you're bringing non-Cal friends, make a reservation online ahead of time and it costs ~$14 per person (student discount with student ID at another school). There is limited parking by the garden, but there is a shuttle that runs from campus up to the garden.
Highlights
The arid house has amazing succulents from around the world.
The rose garden has great views of the Bay.
A great selection of California native species.
Events
The garden has plant sales twice a year. You can get unusual plants for very reasonable prices. If you're very keen on rare plants, it's worth getting a garden membership so you can go the day before the sale is open to the public.
There is a summer concert series in the Redwood Grove featuring a variety of genres.