Itinerary Suggestions France/Germany

suggested help with transportation train vs. car rental would be a big
help.

In general, train should be your default option. But it depends on where you want to go.

Is it Hamlin or Hamelin?

Here are my comments about possible destinations in southern France from your earlier thread, which focused more on transatlantic transportation:

"As for staying in Perpignan, there may be two days' worth of sights in that city; after that, you'll probably be looking at day-trips. To the south you have Collioure (French beach village) and Figueres (Dali Theatre/Museum, in Spain). To the north you have Narbonne (attractive French city). To the Northwest you have Carcassonne. To the west you have the Yellow Train trip through the Pyrenees from Villefranche-de-Conflent to Latour de Carol; I don't know that the full trip could be made in one day, especially in the winter when the train schedule may be more sporadic. To the southwest you have the attractive French town of Ceret.

"I don't travel to Europe as early in the year as you're planning and haven't checked the weather conditions at that time of year. You might do better to focus on major cities where there will be more indoor activities. Even aside from the weather issue, Perpignan is a rather unusual choice as a base. I liked it well enough, but I chose it mostly because it was convenient for day-trips."

You can use the French rail website to check on whether places you're interested in will be viable day-trips from Perpignan. I often start with the German rail website instead, because it covers most of the trains in Europe and I find it easier to use. I've sometimes found trains missing there, though (and it probably won't give you fares for domestic French routes), so be prepared to check both sources for the French part of your trip.

If you don't find trains to somewhere you want to go, use Rome2Rio to see what sort of bus service might be available. However, Rome2Rio's information on fares, travel times and frequencies absolutely cannot be trusted. You need to keep drilling down until you reach the name of the company operating the buses and go to its website for the information you need.

To decide between train and rental car you must first figure out where you want to go. Renting a car in France and dropping it off in Germany would almost certainly incur a large international drop charge.

For short-distance day-trips on regional (not express) trains, the fares are usually very inexpensive. The same is true of bus fares. Express trains (TGVs in France) can be much more expensive if you buy the tickets at the last minute. You'd need those if you wanted to try taking some rather long-distance day-trips. For example, the TGV (or Spanish AVE) will get you from Perpignan to Barcelona in as little as 1 hr. 22 min. ViaMichelin estimates the driving time at 2 hr. 4 min. without any stopping, getting lost, traffic tie-ups, looking for parking, etc. If you contemplate using a rental car to drive to a large city, keep in mind that parking could turn out to be very expensive, on top of the car-rental cost and the fuel (much more expensive in Europe than in the US).

For estimating driving times, use ViaMichelin.com.

For a wealth of general information on the trains in France and Germany (and elsewhere), go to Seat61.com.

You can fly from the US to Hannover that includes a connection in Europe from: Atlanta, Austin, Boston, Charlotte, Chicago, Cincinnati, Dallas, Denver, Detroit, Durham, Houston, Las Vegas, Los Angeles, Miami, Minneapolis, Nashville, New York, Orlando, Philadelphia, Phoenix, Portland, Salt Lake City, San Diego, San Francisco, San Jose, Seattle, Tampa and Washington DC.
I recommend going to DE first because there’s a direct train from Kaiserslautern to Paris (2h 45m) but not the other way around.
From Hannover you can take a direct train to Hamelin (45m). From Hamelin you can take a train to Kaiserslautern switching lines in Hildesheim (1h) and Mannheim (4h 15m) and arrive in Kaiserslautern 1h later. I would sleep there and take a day trip to Landstuhl (15m).
From Paris you can take a direct train to Perpignan (5h). I would then take the train back to Paris (5h 15m) and fly home from there. You can fly nonstop from Paris to: Atlanta, Boston, Chicago, Cincinnati, Dallas, Detroit, Durham, Houston, Los Angeles, Miami, Minneapolis, New York, Philadelphia, Salt Lake City, San Francisco, Seattle and Washington DC.
Good day trips by train from Perpignan is Collioure (30m), Carcassonne (1h 45) and Figueres to visit Dali's Museum (30m).

It may be necessary to wait a while to check on the viability of day-trips. At the moment I'm not seeing any morning TGV options from Perpignan to Figueres. That's a surprise and may be entirely due to COVID-19 travel restrictions.

I recommend going to DE first because there’s a direct train from
Kaiserslautern to Paris (2h 45m) but not the other way around.

There are, in normal times, direct trains from Paris to Kaiserslautern.

"Is it Hamlin or Hamelin?" This nice old town on the river Weser is called Hameln in German. This is the important expression you should use when ordering online tickets or using ticket machines at train stations. The town is mainly known for the fairy tale by the brothers Grimm 'Der Rattenfänger von Hameln' (literal translation: The rat catcher of Hameln), known in the English speaking world as
'The Pied Piper of Hamelin'.

In general, train should be your default option

Generally true for Germany, which has the largest rail system in Western Europe and the second most dense (by 1.2%) to Switzerland's. I've traveled extensively in Germany, mostly to small towns, without ever needing a car.

Not so much in France; with almost twice the area of Germany, France has only 70% as much trackage. There are large parts of France inaccessible by rail.

If you decide to use a car, a couple of companies ( Peugeot, Renault ? ) have lease programs that they say are appropriate for 3 weeks or more usage.

Germany is very easy to travel by trains to most of the usual towns that Americans are likely to visit. France much less so. Of course you can rent a car to use for trips not easy to make by train.

acraven - DB shows direct train service: https://reiseauskunft.bahn.de/bin/query.exe/en?revia=yes&existOptimizePrice-deactivated=1&country=GBR&dbkanal_007=L04_S02_D002_KIN0059_qf-bahn-svb-kl2_lz03&start=1&protocol=https%3A&REQ0JourneyStopsS0A=1&S=Perpignan&REQ0JourneyStopsSID=A%3D1%40O%3DPerpignan%40X%3D2879398%40Y%3D42695939%40U%3D80%40L%3D008700164%40B%3D1%40p%3D1616702960%40&Z=Figueres+Vilafant&REQ0JourneyStopsZID=&date=Thu%2C+08.04.21&time=11%3A00&timesel=depart&returnDate=&returnTime=&returnTimesel=depart&optimize=0&auskunft_travelers_number=1&tariffTravellerType.1=E&tariffTravellerReductionClass.1=0&tariffClass=2&rtMode=DB-HYBRID&externRequest=yes&HWAI=JS%21js%3Dyes%21ajax%3Dyes%21#hfsseq1|mi.0287382.1617324841