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There's two good resources:

• Formula Operators and Functions
• The Build Formula insert actions within the Process Builder (see below)

By searching through the functions and clicking insert, you'll see the format for functions.

There's two things within your example that is off:

1. Parenthesis - if you want all conditions met, make sure they're all within the AND(..everything here...)
2. Your original formula used CONTAINS correctly (but your additions did not) and the same format/syntax applies even if using a field instead of a custom metadata value.

CONTAINS should be CONTAINS(text, compare_text) . If that text is a field, then it's just CONTAINS(fieldName, compare_text).

AND(
$CustomMetadata.Process_Builder_Bypass__mdt.Email_Auto_Update_Provided_To_Customer.Bypassed__c,
CONTAINS($CustomMetadata.Process_Builder_Bypass__mdt.Email_Auto_Update_Provided_To_Customer.Profiles__c, $Profile.Name),
CONTAINS([EmailMessage].Parent.Status, "TR/CR"),
CONTAINS([EmailMessage].Subject, "_TR_CR_")
)