check the airline website for the rules on hand luggage weight and size. If weight is an issue, DON'T use a bag with wheels, they are dead weight. If it is too heavy to carry, take less.
she MAY be allowed a handbag as well as her cabin bag, again depends on the airline.
the plastic bag is for liquids, gels and pastes - so all makeup, toothpaste etc. If she gets travel sizes and is sensible on the slap, it will all fit in the bag. No container can be over 100ml, end of. These amounts are plenty for a week for everything except suncream - for that, take a small container and then buy abroad, NOT on the plane which is crazy expensive.
packing list for a week somewhere warm-ish: - four t-shirts, one swimsuit, two pairs shorts, one skirt, two dressy tops, underwear, one change of shoes. She travels in trousers, trainers, a couple of t-shirts and a fleece top, plus a rain jacket if she will need that.Also the obvious; passport, insurance, money, cards, book, MP3 and medication. Yes, I know she's a teen but that doesn't mean she HAS to be high maintenance.
top tip - take an empty water bottle through security, then fill from the tap afterwards.