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!"
seriously, Mo Willems is the modern-day Dr Seuss, it's much better than what we grew up with!
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
Ursula Vernon (especially the Hamster Princess series) - Vernon writes quirky, funny books including the Hamster Princess series, which is a funny funhouse-mirror version of Sleeping Beauty (some of the later books in the series aren't as good as the first ones, but they're all fun). Adam also enjoyed her other books like Dragonbreath and Castle Hangnail.
Popular, I don't whole-heartedly recommend, but aren't bad:
Magic Treehouse series - these are very formulaic but relatively easy for a kid who's just starting with chapter books, and each one teaches you about a different time/place
Captain Underpants series - these are all full of cheesy jokes and bathroom humor, but they are like catnip for certain kids, and they do have some fun self-referential humor; the same author also wrote the Dogman series but Adam never got into them because cats are the villains and he loves cats
Mercy Watson series - these are illustrated in an old-fashioned, 1950s style, but are full of humor
...also consider graphic novels, which are a great way for early readers to get into reading; Adam got obsessed with a bunch of Minecraft-themed ones