Which Do You Read First: Survey
Generally, private messages first. If it's about something in the game, I will respond to acknowledge I got the message, and state that I'll clarify further after I've read the public stuff.
As far as the public messages go...I usually end up reading them in order of oldest to newest threads showing new messages, regardless of whether they're OOC or IC threads. And, most of the time (but not always), I'll read everything before responding to anything. If the only threads showing new posts are IC threads that my character is involved in, I'll post as soon as I've read, because I'm generally pretty sure nothing in any of the other threads is going to impact what I'd post...I'm not always right, and sometimes I'll go back and edit a post accordingly, or write another post, but that happens rarely.