Having attended four landscape painting courses in France, I have carried just enough, too little, and too much. I have carried my painting supplies to lots of other places too and I'm interested in learning from others.
My traveling easel of choice was the Soltek for France. Provence has lovely weather but it's surprisingly windy. One of my classmates had a painting fly away into a ravine and others had their whole French easels tip over. The Soltek stayed secure and held the painting surface well. It also has a handy place to hold the palette and other necessities. It weighs about 11 lbs, probably more if you buy the tall person version, and it's best to pack it in your checked luggage I think. I like to fill its trays with brushes, rags, charcoal, sharpeners, viewfinders etc. I pack paints separately. The only caution with Soltek is to keep the locking mechanism on the legs clean. I carry three plastic bags to put over the ends of the legs if I'm in mud or sand. The pallette I use came with the Soltek. It's grey plastic, easily cleaned and just the right size. I own two and alternate between them.
I have also traveled with a cheap aluminum easel as hand luggage and it's quite OK if weight or space is an issue but be sure to lock the legs securely or even dig them in if you are in a windy place and be sure your canvas is nice and snug.
You can't bring turpentine or other solvents on a plane so I buy mine when I get there. If there isn't an art store, I go to hardware stores and pick up white spirit or turpentine. In France I found white spirit in one of the big supermarkets. In Scotland I searched online in advance and found an art supply shop close to my hotel. My friends in Arizona keep a can in their garage for my annual visit. Another solution is to bring with you a product called Spectrum clear that can be used for painting although not for cleaning brushes. It is a synthetic like Liquin but I don't like the finish it gives so it's for emergencies only.
Paint colors I like include a basic minimal palette, white, burnt umber, cadmium red medium, alizarin Crimson, cadmium yellow medium and French ultramarine. I can usually mix anything...at least naturally occurring color with those but to save time I carry some other favorites too, such as burnt sienna, yellow ochre, Kings blue, cobalt blue, cerulean, lemon yellow, Mars orange, Jaune brilliant and a variety of greens.
I pack half empty tubes each in its own ziplock bag. New tubes can burst with the pressure changes in a plane. I pack the lot in a plastic art bin and carefully label them for security. Here's a tip. They are oil colors, not oil paints. Oil paints for TSA purposes means oil based house paints so are prohibited from planes. Artists oil colors are safe to carry so put a notice with them that says artists oil colors containing vegetable oils and I note that I am not carrying solvents. I also include a printout from the TSA website which says they are allowed and a material safety data sheet from one of the manufacturers that has the flash point. They are actually legal as hand luggage too but then you have to go through the whole liquids thing so I never try that.
I mostly use prepared paper while traveling so I pack a piece of foam core to tape it on but cardboard is probably just as good. I sometimes bring panels but I find stretched canvas just takes up too much space in my bags and boards are too heavy.
So other things I pack include baby wipes for cleanup, some plastic bags for rubbish, artists tape, spice jars for solvent, a scissors and pallette knives ( not in hand luggage of course) and for drawings, a journal and soft graphite pencil and a kneaded eraser. I generally can find paper napkins just about anywhere so I don't bring paper towels but I do pack a roll of plastic wrap to cover my pallette while it's wet and a few sheets of wax paper to put between still tacky paintings. I have a huge reusable bag from Lidl that I pack in my checked luggage. Once I get to my destination I put everything except the easel in there and I'm ready to roll.
I find using white spirit is less stinky than turpentine and dries enough in a few days to carry paintings home safely but I often mail home finished paintings and anything else I want to send home after I stop painting. I stop painting two or three days before flying back so I won't have wet paintings but a fan can speed up drying if you can find one. The back window of a hot car dries paintings fast too. Friends have tried hair dryers but that's a bit tedious.
Don't forget a sun hat, sunscreen and insect repellent and be sure to bring water and a snack with you. It's easy to lose track of time and get dehydrated or sunburned or suddenly realize you are starving. Happy travels!