There really isn't anywhere to see the contract date online, though you can tell when it started by looking at your prior invoices on the My Bill tab of the HughesNet My Account site. If your contract was as the result of an upgrade, you can sometimes tell by when the monthly price changed with those invoices. Also, you can see what plan you have by signing into the same site, hovering over the My Services header, and clicking on the My Service Plan tab.
To be sure, you should call HughesNet Customer Support at 866-347-3292 and ask them. In order to cancel you have to call that number.
Hope this helps.
Edit: I should add that, depending on the price of your plan, if you only have a few months left on your contract the Early Termination Fee can actually be around the same as, or even less than, your normal monthly bill. The Early Termination Fee is $400 for the first 90 days, then $15 less per month thereafter, so if you only had two months left, as an example, your ETF would be somewhere around $115 ($400 - $285 (19 months x $15)). Sometimes people prefer to just pay the ETF instead of keeping the service for the remainder of the contract time.