Yes, all weightlifting is sanctioned by USAW - there is only one US federation. In order to compete, you do need to be a USAW member, which you can become on their website here
Like Geoffrey said, you don't have to be a member of a club - you can lift unattached.
One way to find meets is to find your LWC
and get in touch with them about meet schedules. Many of them will keep their own websites with meet schedules or in some way publish the info.
You can also find meets on the USAW site here
and sort by state/region. Your LWC might have meet info up sooner than USAW.
According to USAW rules, you don't need a singlet for your first meet, but some meet directors require it, so make sure you check the entry form. It's smart to just get one anyway - you can lift in shorts and T-shirt, but depending on the judges/director, this can wind up being more work than just getting a singlet.