I typically use a Mac and in MAC Mail I can sort my emails by subject and move many of them at the same time to saved folders with client subjects to appropriate folders. Is there a way to do that with Gmail?
Of course there is. However, Gmail orders messages (conversations) by date; you cannot change the sort order to be by subject.
What you'll want to do is filter your messages. You can specifically target the subject by using the
subject: search operator.
subject:"You're a winner"
That search will find all messages with "You're a winner" in the subject line. You can then easily select all or some of them and change the labels as you need. (Note that Gmail uses labels, not folders. Messages/conversations can have multiple different labels.)
If you know what the subject (or part of it) is going to be ahead of time, you can set up filters to do a number of things on the messages as they come in, including applying labels. The basic way to create a filter is to search for your criteria, then "create a filter" right from the search drop-down. Choose your actions and you're done.
More information on this and much more can be found at https://support.google.com/mail/