Visiting France and Germany

Airlines only book 330 days out, so in a couple months you can start researching airlines. You can use google flights or matrix.itasoftware.com.
Offhand, there are nonstop flights to Paris out of Miami on American and Air France. Later on, you may also find nonstops out of Orlando. As such, you might want to start with Paris, then travel to Perignan by train. Then maybe a discount airline like Easyjet to Germany. Fly back home from Frankfurt. Use the multicity search function to book open jaw.

There are other options, but I do my flight research on Google Flights and then book my flights on the appropriate airline's website. You'll be looking for a "multi-city" flight into your first destination (or the major airport near it) and home from your last (or a nearby airport). You are not looking for two separate tickets. The more options you give yourself for departure and arrival airports, the better your chances of not paying a lot for the flights. Edited to add: To clarify, I use Google Flights for transatlantic flights but agree with Mary Pat that Skyscanner.com is good within Europe.

Perpignan appears to have an airport, but it must be very small. You may do better to head for one of the larger airports not too far away and then take a train. The nearest major airports are Barcelona (Spain) and Toulouse (France). Marseille (France) isn't terribly far, either. From my origin it would almost certainly be a lot cheaper to fly into Barcelona, but your situation may be quite different.

On the other end, Landstuhl is in the middle of a triangle between Frankfurt, Luxembourg and Strasbourg. Icelandair used to fly into Luxembourg; I'm not sure it still does. Otherwise, Frankfurt is likely to be cheapest.

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 take a look at actual, day-by-day, historical weather statistics on timeanddate.com. If weather matters to you at all, be sure to review at least 3 years' worth of data; 5 years would be better.

Sam gave you good advice, but I prefer Skyscanner as my search engine. Air France does have a flight that departs from Miami and arrives in Paris where you can continue on to Perpignan. You can then take a flight from Perpignan to Hamelin but will need to change planes in Paris.
As far as getting around in a country, train travel is best for those who don’t want to rent a car. Of course, that can be cumbersome and a car will allow you to get off the beaten path.

You could look also into flying internationally into Barcelona and then back from Frankfurt and take the train between destinations.

As far as getting around in a country, train travel is best for those
who don’t want to rent a car.

The train is often the best way to get around, even for those who want to rent a car. The TGV network in France is very good. The drawback of it is that is very Paris centric, so if you are not going to or from Paris it is a bit worse. Germany on the other hand is much more polycentric and that is reflected in the railway network.

For the long-haul portion of your itinerary, for departure airports check Miami, Atlanta, or even Charlotte. For my wife and I, if we’re traveling around a country to many smaller or out of the way places, we’ll get a rental car. Driving in France and Germany is the same as the U.S.. No need for a car in large cities where getting around via public transportation is easy and parking is a pain and expensive. Most parking anyplace other than really small towns and villages, is pay parking. Finally, speed cameras are everywhere so don’t drive over the limit. Ultimately, which places you decide to visit should give you and idea as to whether a rental car or public transportation is best for you.

The closest airport with frequent service is Montpellier (MPL). To reach it from Vero Beach, I should think that by February or March there might be some options from either MCO or MIA to an intermediate airport such as CDG, LHR, AMS, or FRA and connect from there to MPL.

From MPL, you can take a taxi, about 20€ to the train station in Montpellier from where there are typically a number of trains to Perpignan.

The closest airport with frequent service is Montpellier (MPL).

I guess that depends on how you define frequent service. Toulouse and Barcelona are only slightly further away and both offer a much larger selection of flights.

Both TLS and BCN are over 50km farther from Perpignan than is MPL. For each of these choices, getting from the airport to the train station is far easier in Montpellier. Getting to the Toulouse train station from TLS is a pain. There is TGV service from Montpellier, and possibly from Barcelona, but service from Toulouse eastward is not all TGVs and may require a connection at Narbonne. I would not personally recommend TLS for someone going to Perpignan.

But we should consider that Patty is coming from Florida. She might be able to find a MIA to BCN flight and make a reasonable train connection to Perpignan. But both MCO and MIA normally have options to CDG/LHR with connections to MPL Tickets are typically cheaper when making an en route connection such as via CDG/LHR/AMS vs. a BCN nonstop but with current air carrier service being what it is, there is no way to confirm, or even estimate, what might be the best option.

Both TLS and BCN are over 50km farther from Perpignan than is MPL.

Distance is not everything. If you look at travel time, Barcelona is closer than Montpellier, 1:20 vs 1:40 by direct train. I agree that Toulouse is an outlier but if it was me I'd keep it in mind in case I find a connection where the price and/or time suits me. Barcelona has local trains from the airport to the central station, so it is pretty easy to get to the train station.

I don't understand the american obsession of talking in airport codes but a big advantage of Barcelona is (as I think you mention) direct transatlantic flights. Montpellier requires a connection somewhere in Europe. And even if the OP don't mind a connection, there are far more options to Barcelona as it is one of the busiest airports in Europe, while Montpellier is a small regional airport.

I'm all for considering Montpellier too, but in my experience the cost of flying from a major origin airport into a small airport like that is often higher by $250 to $500 on a round trip (so about half that as part of an open-jaw flight) than landing in a major city like Barcelona or Paris. It's possible the pattern is different if you need to begin and end your trip at a regional airport. It doesn't take long on Google Flights to figure out that sort of thing.

True, Patty's situation is a bit more complex because she also has multiple origin-airport options. She might want to start by entering each of her possible origin airports (one at a time) on Google flights and "Europe" as the destination, then clicking on "Explore destinations" in small blue text above the map. Zooming in on the map will bring up more and more round-trip fares for varying dates. You can tweak the duration of the trip in a box to the left of the map; I just used 2 weeks in March 2021. In normal times (which these are not), travel dates can significantly affect fares but often don't make a major difference in the relationships among fares to different destinations.

Patty won't be flying round trip unless she finds Paris to be markedly cheaper, I assume, but if the RT is a lot more expensive to Airport A as opposed to Airport B, the multi-city trip using Airport A in just one direction will probably be more costly by about half that amount. This is what I found using the "Explore" function with MIA as the origin:

Barcelona $445
Paris $480
Toulouse $615
Marseille $772
Montpellier $2,114

Since the flight home will not be from southern France/northern Spain, I also looked at RT fares to the area around Landstuhl:

Luxembourg $564 on March 28; more typically $900+
Frankfurt $610
Zurich $606 (rail fare to Zurich might be higher than to the other cities)
Strasbourg $2445

Luxembourg can be a good option if Icelandair flies to an airport near you.

Google Flights' Explore function is producing some odd-looking results for me at the moment (perhaps because many routes just have no service now?), so I'd recommend confirming results with standard searches that specify the origin/destination pairs you want to explore more fully. It's quite possible that Strasbourg and Montpellier won't look so wildly off-base by fall, but I bet they'll still be unattractively priced from MIA by comparison to the other possibilities. The results for Orlando and Tampa might be different.

I have to ask, why Perpignan? I spent a week at a friend’s timeshare in Canet-en-Roussillon. It’s on the coast about 7 miles west of Perpignan. We spent the week with some beach time (late September/early October) and day trips to a variety of places. Sadly, we didn’t explore Perpignan because my friend wanted to see the other places. We had a car. Depending on where you want to go it would be possible to use train travel exclusively. We spent a few days in Barcelona before taking the train to Perpignan. We trained to Paris from there.