At first, there will be a learning curve to see what the client deems important/not-important. Once you get through that, use the Gmail labels (and sub-labels) copiously. For example, if there are action items for the client, you can label those 'Action items' and have a sub-label called 'resolved' or completed, or something to that effect.
Ask your client which labels they already have that are off limits to you. While you have access to everything, a client may want to have some personal or sensitive info in a sub-folder. Just by asking a client about this, will allow to the client to see that you respect their privacy and that will build trust.
If the client is a solopreneur, take the time to think ahead about other labels (you can think about how you set up email and the labels or folders that you have as a guide). A label for "web stuff" like domain registration information, expiration dates, logins, etc., or a label for electronic receipts (like when you make a purchase on Amazon, FTD, etc.) are handy. The receipts label is good so that you can also get an idea of what the client has purchased in the past should you need to reorder in the future.
Finally, make less work for everyone and set up filters for everything you can. Makes life easier