Since carrots and other veggies don't have seeds, how do they grow from the ground using seeds? Secondly, how do they get the seeds to grown them?

The seeds are not in the edible part of the carrot but grow from the flowers that form from the greenery above.

In other words, Carrots, Onions and potatoes are the roots of the flowers. We are eating the roots of the plant rather than the plant itself. As an experiment cut the top off a carrot and place the carrot in a cup of water and watch the top grow.

Potatoes, if you let them grow, will grow up similar to a tomato plant, and will actually produce fruit which are like small green tomatoes, although I don't think they are edible.

Vegetables can reproduce in two different ways - by producing flowers and seeds, or by making copies of themselves directly. This latter is the normal way. The new copies are clones of the original plant rather than a cross of a father and a mother.

All parts of the potato plant apart from the tubers (swollen roots) are poisonous. And any green parts on the tuber are not to be eaten, either. All plants that you will come across normally grow from seeds, with a few exceptions. Bananas don't produce seed that will grow any more, so they are grown from cuttings of existing plants - and the same goes for seedless grapes etc. Mushrooms and other fungi, and ferns, grow from spores which are a bit like seeds in a way. Fungi don't flower, nor do ferns. The seedless fruits do flower - or there wouldn't be any bananas - but the seed is either not formed or is not growable. Gnomon, I would say the normal way for veg to reproduce was seed. Cloning - which is what happens when a potato plant grows from each tuber rqather than just one rot as in the carrot - is less common than seed production, and all the self-cloning plants do produce seed as well.