There is plenty to do in northern Germany, so I wouldn't feel compelled to fly into or out of a different area or country unless there was a really significant financial advantage or I had a burning desire to include such other location in this itinerary.
You have to be careful about choosing distant gateways for purely financial reasons. It's ill-advised to book onward flights or non-refundable trains in advance for the day of your arrival since delays are possible. Therefore, you're probably going to pay full-fare for that onward transportation or need to spend the night at your arrival city. And at the end of the trip you'll usually burn at least half a day (more if flying) to get back to your departure point if it is other than the last city on your itinerary. This means the perceived financial advantage of grabbing a cheaper flight may not be real.
I'd start by visiting a flight-search website; I use Google Flights but there are other options. Start with Columbus, but do check the other viable origins. Last fall someone from Cincinnati stumbled on give-away fares to Europe, so you just never know.
See what it will cost to fly into and out of any combination of Bremen, Hamburg, and maybe Berlin. (To mix and match cities, choose the "multi-city" option.) I hear you about Berlin; love the place, worth a lot of time. But it's only about 2 hours by train from Berlin to Hamburg, and I see advance-purchase, non-refundable rail fares as low as 19 euros for June 2 (date chosen randomly). For the right fare differential, into or out of Berlin might make sense. For the right financial pay-off you could spend the first night or two in the Berlin area and just do something like see Potsdam. Then on Day 2 or Day 3 you'd use your cheapo advance-purchase train tickets to get to Hamburg, Bremen, etc.
I haven't seen much at all of northern Germany (others here have and will surely come along and give you a bunch of good ideas), but I do have one suggestion: Schwerin. It's a very pretty historic town with a castle perched in a man-made lake. Worth about a full day if you want a relaxing visit, and it would be a good place to decompress after a long flight. It's less than 2 hours from the Hamburg airport. I'm not sure what same-day rail tickets would cost. Tickets for tomorrow are available at 19 euros.
Edited to add: You can explore train schedules and fares on the Deutsche Bahn website. Is schedule for your travel period aren't available yet, choose the same day of the week as far out as the schedule will go to see advance-purchase rail fares.
Edit #2:
I've just played around a bit with Google Flights. You can give it an origin airport and choose "Germany" as your destination. Then plug in your dates. I didn't have specifics, so I checked for a 2-week period in June and a 2-week period in July. When you get down to specifics, things may look different, but my quick research suggests (note caveat; you need to replicate with actual travel dates):
Pittsburgh is probably your cheapest origin. Detroit is next cheapest. Forget about Ohio if you want to fly into/out of Germany.
Cheapest gateway in northern Germany varies with your US origin. From Pittsburgh, it looks like Frankfurt - Hamburg - Berlin. From Detroit, it's Hamburg.
June is likely to be cheaper than July, but there's a great deal of variation by date within some months for some origin/destination combinations.
I did not look at flight duration/number of connections.
I set "Europe" as the destination in June from each potential origin and didn't find anything that looked enough better than the German cities out of Pittsburgh to be worth the hassle--perhaps $100 cheaper into Amsterdam. But if the flight times are a lot better...
I did not try multi-city flights (into/out of different cities).