Thanks again for all your help. The scripts I am using sometimes specify a valid From address, so that may be the problem. If the user filling out the web form includes their email address (optional), the from field is populated in the headers with their email address. Otherwise, it is blank, which is probably going into the Junk Mail. I think that isolates the problem.
If the SMTP alternative is involved, I'll probably continue to just use mail(). I at least can get mail() to work ok, and I'm not up for hours of troubleshooting right now. I'm not that skilled with these things, and sometimes still struggle with some of the PHP basics. At least I know now the importance of specifying a valid From address to avoid being considered Junk Mail.
Thanks,
Troy