Get a library card!
There are so many books at the library—not only is it more economical than buying a bunch of books, but it also gives you a chance to "try before you buy". San Francisco's Public Library system is awesome, with convenient neighborhood branches, no overdue fines (!), and relatively easy holds/interlibrary loans. You might wait until your kid is ~1 year old or so, since babies tend to chew books...
Board Books for Babies/Young Toddlers
Before your baby is about 2 years old, you'll want to mostly get "board books", which are harder to destroy
Goodnight, Moon - classic
Sandra Boynton - lots of fun books with fun rhymes and drawings (including Boynton's signature cross-eyed animals). Among our favorites were Barnyard Dance, Moo Baa La La La, Blue Hat Green Hat, Night Night Pookie, and Oh My Oh My Oh Dinosaurs.
Eric Carle - Hungry Hungry Caterpillar and Brown Bear are classics for a good reason (although we didn't much care for all the sequels to these two)
One Gorilla - nice counting book (FYI ~all the negative reviews on Amazon are from anti-evolution creationists!)
Books for 2-4 Year Olds
Here are some books we liked:
Jeffrey Brown's Star Wars books - "Vader and Son" ended up being one of Adam's favorite books!
I Don't Want to Be a Frog - cute/funny book about a frog who wants to be something else
Click Clack Moo - a farmer's cows learn to use a typewriter and hilarity ensues
Dragons Love Tacos - funny book about dragons—don't give them spicy salsa!
In My Heart - a beautiful book that also helps toddlers learn about emotions (note: we did not think the sequel "All My Treasures" was nearly as good)
Interrupting Chicken - a fun "meta" book about bedtime stories; best if your child knows the story of Hansel & Gretel and Red Riding Hood already. Also has a sequel (which isn't bad, but is probably funnier for adults than kids)
The Book with No Pictures - written by BJ Novak (actor/screenwriter), this book indeed has no pictures but is a lot of fun for both toddlers & parents to read
You Don't Want a Unicorn - similar to the "dragons love tacos" book, shows how unicorns can be impractical
Pigeon series by Mo Willems - a bunch of funny books with a pigeon who acts a bit like a toddler
our favorites so far are The Pigeon Finds a Hot Dog! and The Pigeon Wants a Puppy!
Elephant & Piggie series by Mo Willems - designed as "early readers" (for kids just beginning to read), these are nearly all hilarious. Some of our favorites include "There is a Bird on Your Head!" and "I Broke My Trunk!"
The Gruffalo - a fun, rhyming book about a mouse who successfully outwits his foes; there is a really nice animated ~half hour version on Amazon Prime starring a bunch of famous British voice actors including James Corden as the mouse.
[if you have a truck fan: Digger, Dozer, Dumper - somehow Adam got obsessed with trucks, and this rhyming book is the best of the truck-centric books we found]
Read Aloud Chapter Books (4+ Year Olds)
Early school-aged kids are just starting to read on their own, but love hearing more complex stories when read to.
Harry Potter series - we were surprised how much Adam loves listening to the Harry Potter books; they're also available as audiobooks if you don't want to read yourself :-)
The Little Prince - this is an old classic that Adam keeps wanting to hear again and again; it's surprisingly deep