You definitely need to check out specific airlines as not all carry on dimensions / weights are the same. Some allow a second smaller bag eg handbag too.
How long are you going for? I've successfully managed 3 night breaks with a relatively small rucksack. Easier when the weather is predictable and you're not having to cater for varying weather conditions.
As others have said, find out if your accommodation provides any toiletries. I always take conditioner because it's not always provided. Usually taken from a hotel on other trips where I've had a bigger suitcase.
I ask my dentist for toothpaste samples, they almost always have them.
I wear layers on the plane usually a fleece plus coat or jacket and travel in my bulkiest garments eg jeans and trainers or walking shoes. But the layer closest to my skin (eg T-shirt) I make sure is quick dry material. If I'm there for 3 nights, I wash the base layer and underwear I've travelled in. it will dry for travelling home. Shampoo or shower gel are effective for hand washing
Roll clothes tightly or use compressed packing cubes that fit in your case//rucksack efficiently.
Keep to one colour palette so you don't need different shoes/jackets.
One neutral eyeshadow, one eyeliner, one mascara and one lipstick/lipgloss. Decant a small amount of foundation but I generally don't bother with that. If you do like to wear foundation, take a lipstick shade you can use as blusher too.