Advice on where to start and end in Europe

Start wherever it’s easiest to fly into from wherever your home airport is.

Fly open jaw.

Work out where you want to go in the south of France, Spain and Portugal as this will affect your route.

Use rome2rio.com to get an idea of travel options. You will most likely be flying between all your destinations, as they cover a wide geographical area. I hope the trip is at least 3 weeks long to warrant so much time in transit.

How long is the trip and where in Spain and Portugal do you plan to go. Can't really give decent advice without knowing those things. But as said above, the distances are far so most likely you'll need to fly unless you have a lot of time (3-4 weeks at least) and plan to see other things between them. In that case you could train. Driving is not really practical because picking up a car in one country and returning it in another will incur huge drop off fees (hundreds).

How long do you have? It looks like at least a month long trip or more.
Fly into one, and out of the last place on your itinerary.
You can do a mix of driving, flying, trains and buses. I find spending all of my time on a trip in one country is much more rewarding.

Fishbowl, without a LOT more info, it's pretty much impossible to respond to your question. We need to know stuff like:

Who is "we"? Two people? More than two people? Ages? Any mobility considerations?

How much time do you have for this trip? What time of year?

Bus/train/car? How comfortable are you driving abroad? How much effort are you willing to invest into, say, researching Italian traffic laws?

Aside from Lucca, Italy, where in Spain and Portugal do you want to visit? Where do you intend to fly into Italy to go to Lucca (which doesn't have an airport)? From where in the world are you coming from?

You're talking about a trip that covers at least 1,800km or about 1,200 miles through 4 countries, a very large area, which could require multiple weeks to visit without rushing. Best transportation options will depend on your actual itinerary, but the suggestion to fly open jaws (e.g. into Lisbon and out of Pisa or vice versa) is good advice. Time of year can also be a factor. At the moment, your question is too general to answer more specifically.

Such an itinerary just doesn't flow very well. You'd do best to just visit Spain and Portugal and leave the other places to your next trip.
We try to visit cities that are in the same geographic area, cities that compliment each other with good train connections between the cities. Our travel is so much efficient that way.

Lucca is served by train, so you can get there from almost anywhere in the country, but most likely airports would include Pisa, Genoa, Milan, or Rome. Trains along the Italian Riviera and connecting to France are not particularly fast. If you don't have other destinations planned in Italy, then I'd probably cut Lucca and fly into Nice or Marseille for the south of France.

Looking Up Train Schedules and Routes Online gives you the Deutsche Bahn train schedule link and tips for using it. Rick’s Train Travel Time & cost Map gives you an overview of faster train travel times in hours, as well as regular (full-price) 2nd-class fares. If some travel times look too long, you might fly, particularly between countries.

It your destinations are far apart or poorly-connected via ground transportation (e.g., Spain and Portugal), you'll need flights unless you have a great deal of time indeed. Once it's necessary to fly, it doesn't make a great deal of difference in what order you visit the various countries. I'd start by looking at airfare from your origin to each potential destination and from each other potential destination back home. Some destinations may be cheaper than others, but it varies greatly by origin. For me Paris is cheaper than other places in France but Nice is often not too much more expensive. In Italy, Milan and Rome are usually cheaper than the other options. Barcelona and Madrid are virtually always the cheapest destinations in Spain. I haven't priced flights to Portugal recently but would rather expect Lisbon to be the cheapest. What I don't know is how Lisbon, Barcelona/Madrid, Paris/Nice and Milan/Rome compare to each other.

I suggest exploring on Google Flights. I always start by checking round-trip fares even though I ultimately want to buy a multi-city ticket (into one city and home from another). The round-trip fares are helpful in directing me to the least expensive workable European gateways.