For a start, need more info to help:
Email/contacts provider
Account setup type on phone
Contact sync settings in iTunes
Deep down in your Mail, Contacts, and Calendar settings, there is an obscure option for Contacts - Default Account. Change it from On My iPhone to the name that you gave your Exchange account. Then it will add them to Exchange instead of just the local list. And like previous posters said, disable iTunes from syncing this, otherwise you'll get duplicates.
When I add a contact manually on my iPhone am I suppose to add it in the MS Exchange group?
If you're in the group called All Contacts, it'll automatically add it to your exchange account. Or you can be in just the Exchange group too.
Deep down in your Mail, Contacts, and Calendar settings, there is an obscure option for Contacts - Default Account. Change it from On My iPhone to the name that you gave your Exchange account. Then it will add them to Exchange instead of just the local list.
Only way to see if it's in the right group is to go back and forth between the groups to see what's missing. Or browse through your Contacts online and compare the list that way.