In reading message boards and answering student questions, I see many of the same themes come up over and over. People want help locating birth, marriage, and death records for their ancestors... and don't know how to start. Here are a few suggestions:
1) Determine what time period your ancestor lived in: I know it seems simple, but records vary widely depending on time period.
2) Determine what records are available for your time period: In Connecticut, vital records may exist back to the start of the town. However, the state did not require recording until 1897. While there is an index available for some dates before 1897, it does not include every record and is missing most records after 1860.
3) Determine the town's name for the period - and where those records are currently held: Connecticut town names have changed over time. Can you find East Middletown on a map? What about Chatham? You'll need to determine where the town's records were held when they were written. The records haven't moved - but the town's boundaries have. There's a Connecticut State Library webpage that can provide a good way to start.
4) Learn what records might exist for your town. Start with the FamilySearch Wiki for Connecticut for more information.
...And start researching.
Happy Research! If you have questions, please comment below!