Probably the most popular (or at least, the most often cited) store where celebrities go to get ready for their babies is Petit Tresor in LA. They have quite a selection:
Now Katie Holmes and Tom Cruise's attorneys reportedly did have to send Petit Tresor a "cease and desist" letter when they thought store employees were leaking news of purchases they supposedly made to the media, but if you aren't a celebrity, then that shouldn't be a concern for you. This store's popularity is apparent just from browsing the website: in addition to all the merchandise they carry, they offer quite a range, from casual furniture and decorations to much more upscale items.
Another popular boutique for celebrities who are shopping for their kids is Bonpoint. Bonpoint seems much more upscale, so these are the clothes to pick when you want your child to look like she is dressed expensively. Bonpoint's offerings appear to be clothes and shoes, not necessarily furniture, but the clothes are exquisite.
If you want to shop with the celebrities but find more hip/casual things, follow Britney Spears' lead: she reportedly enjoyed shopping at Glamajama for the Mommy and Baby lounge sets. Kate Hudson is said to have liked the "Rock Star" onesies for Ryder when he was a baby. So clearly this store is for the trendy moms! Their items are available via their website and are supposed to be available at JC Penney, so you don't have to travel to Los Angeles to shop there. That's a HUGE bonus for those of us in the heartland, right?
Another option for those of you who don't live near Los Angeles (or New York, for that matter) is Ron Robinson's LifeSize Kids line. It's offered through Fred Segal, but if there isn't a Fred Segal near you, then visit their website. Madonna and Kate Moss reportedly bought clothes for their kids from this line. They offer clothes for infants up through 14 years of age and have baby gear to boot.
So whether you live in LA or just want to dress your child as if you did, there are options, thanks to the internet!