The regulations can feel painful if you are not used to them but most people go through security just fine.
A quart ziplock bag is about 15x20cm. I fly a lot and have never seen anyone measure the bag - so as long as it's about that you will be fine.
Each person in your party can have 1 ziplock bag. Every airport I've been through asks that you have it outside of your carry on luggage for screening.
Prescription medications do not need to be included in that bag but they will be examined so if they fit, put them in the bag. I am pretty sure that all over the counter liquid or gels (or gel tablets) need to go in the ziplock bag. Tablets such as Advil (except the gel kind) can just go in your carry on luggage or purse. I know my daughter's saline solution for her contact lenses needs to go in the bag. She learned the hard way. Aerosols are ok as long as they are no larger than 3 oz (about 85 grams I think).
A few other things that sometimes trip people up - the actual containers that go into the ziplock bag need to be 3 ounces or less, not the contents (in other words, you can't have a half full, 6 oz tube of toothpaste in the ziplock).
Most of the travel sizes sold in pharmacies are no more than 3 oz. You can also get small travel size containers and pour a few ounces of your products in them for carry on in the ziplock bag. I try to travel with a small set of thing I will need for 48 hours. Everything else I put in my checked luggage. If it's in your checked luggage, it is no problem but you need to have enough in your carry on or be ready to purchase what you need if your luggage is delayed.
Things I have seen people try to bring through security that get stopped if they are too big or not in a ziplock bag:
Liquid makeup, snow globes, fruit jelly, lip gloss, perfume, liquid filled vitamins, liquid filled glow sticks, baby's liquid filled teething rings, baby oil.
Not all airports hand out bags and if they do, sometimes they run out so I would bring your own.
When you go through screening if they find something and you didn't put in the bag and it is over 3 oz they will tell you they can throw it out or you can go back and check it. If it is under 3 oz and would fit in the ziplock bag they will probably just tell you to remember the next time and put it in the bag.
There is some variation from airport to airport on what gets through screening and what doesn't, but you don't want your favorite $60, 4 oz jar of face cream dumped in the trash, so take a little time to plan and prepare and you will be fine.
Relax and have a great vacation.
