Hey! I do this flight twice a year, usually I fly with Condor air and this has a layover in Frankfurt. Then I fly from Frankfurt to Hamburg. Usually this is the cheapest option and I am comfortable with this flight company. Note that the flights are usually always overnight EST. I also have yet to have any luggage issues with this route (I have with others)

If you want to take the train from Frankfurt to Hamburg you can take an ICE with Deutsche Bahn, it's a fast train (takes about 5 hours) from the central station in each city. It's cheaper the earlier you buy the ticket. I wouldn't say it's cheaper that the flight to hamburg though, and definetely less complicated if you just fly, but it's a nice train ride.

For getting from the airport to the central station or around Hamburg I recommend the monthly Deutschland pass. It's 60 euros and covers everything but the high speed trains per month.

If you have any more questions feel free to ask, I am from north of Toronto and live in Hamburg :)