The Hartford Line train works well. Note that besides the Amtrak schedule posted by Danill, there is additional train service on this line provided by the State of Connecticut (CTrail). The same ticket can be used on either service (with the special exception of the Amtrak Vermonter). Full details and a combined timetable are at https://www.hartfordline.com/.
The most economical way to get to the train station is probably a local bus; they are operated by CT Transit. Your closest major stop seems to be the Poquonock Park & Ride, just a couple blocks from your hotel.
From the local system map (also linked from https://www.cttransit.com/ but only shows up if your browser window is sufficiently wide), it looks like you could take Routes 32/34/36 to the train station in Windsor.
But another option would be to take Route 30 which runs to downtown Hartford and stops at Hartford Union Station. This will give you more train options, as there are some trains that run only between Hartford and New Haven without continuing to Windsor.
Make sure to check bus schedules because some routes run infrequently or only on weekdays.
You can buy a single trip or day pass on the bus with cash (change will not be given). There is also a contactless Go CT card, which is convenient to use but a bit difficult to obtain except by mail.