The easiest (and most cost effective) way to monitor your router is to use the router's configuration page. Depending on your router (which is information you need to supply when asking router questions), the information provided by the router may be everything you need.
Alternatively, you can purchase router monitoring software. However, this software is usually unnecessary for home networks, and can be bulky to run on a home computer. If you have Linux there are a few free solutions available for network monitoring that might be useful for you. These do use SNMP to retrieve information from your router; however, without knowing which router you have I cannot comment on the availability of SNMP to you.
As far as SNMP goes. It is usually disabled or not even available on home routers because it is more of a corperate tool for large networks. Again, knowing which router you have would make this question easier to answer.