Before you can troubleshoot why outbound mail isn't getting to where it supposed to, you first have to understand how it's supposed to get there. One of the keys to this understanding is knowing whether or not Exchange sends messages directly to recipient systems. When an Exchange server gets a message destined for the outside world, there are two main mechanisms that it consults. The Virtual Server (VS) and Connectors.
The Virtual Server setting takes precedence, so look there first. Open the Exchange System Manager (ESM) and expand to the server in question. Further expand Protocols and SMTP. Beneath SMTP there is typically a single entry - the Default SMTP Virtual Server. Open the Properties of it, go to the Delivery tab, and click on the Advanced button. The main thing you are looking for is whether there is an entry in the Smart host field. If there is, it means that all outbound messages will be sent to that FQDN. If not, your Exchange Organization will require one or more SMTP Connectors to route outbound messages.
If the VS is not configured with a Smarthost, Exchange then looks to the Connectors for outbound routing information. The Connector structure can be simple or complex, depending upon the size of your Exchange Organization and how your enterprise wants mail to flow. For this example, let's assume the simplest configuration possible - a single Exchange server environment.
To route outbound messages, your single-server Exchange Organization requires at least one Connector (assuming you did not configure a smarthost via the VS).
Let's take a moment to review the function and difference between Exchange Administrative Groups (AG) and Exchange Routing Groups (RG). AGs allow you to easily configure administrative permissions to a group of servers, regardless of their geographic location or purpose. RGs are specifically for message routing. There is no implied correlation between any particular AG and RG.
Connectors are associated with RGs and are also referred to as SMTP Connectors.
Getting back our discussion on mail flow, in the simplest configuration you'd have a single Connector. Looking at the Properties, on the General tab there is a radio-button selection allowing you to either configure a smarthost or to use DNS.
If smarthost is selected, there should be an IP address in the field (as opposed to the FQDN format used in the VS).
If DNS is selected, it means that Exchange will attempt to send messages directly to the recipient system.