Ranch 99 is the most familiar Asian grocer in my area. Their inventory includes ingredients related to most Asian cuisines (not just Chinese), as well as live seafood, produce and meats. Their prices are pretty competitive and not unreasonable for hard-to-find ingredients (i.e. Thai basil).
The one thing to keep in mind about Ranch 99 is that customer service is incredibly uneven-- you might be lucky to find an employee who is willing to help you find what you need but most of the time you will be on your own.
Depending on your location parking might be easy to find or a total nightmare (like in my area).