-I don't have a garden per-say, but I do have a place to grow the trees
Outdoors or indoors?
-Unfortunately at this time, I need to spend as little as possible. As for time to find a tree and train it, I have plenty of time
You could try and find good material at a bonsai nursery, or a common nursery - the first one will have better material, but it will be more expensive. Where are you located? if you're lucky, alternative is to get trees from nature, or even from someone's garden.
As much time as necessary I suppose
You will kill trees. That seems inevitable, and a sort of tuition fee. First and most important thing to learn is how to grow a tree - before you try to grow a bonsai.
As for pots, does that mean I should just use any old pot or is there a certain pot I should use to begin with?
You do not need a specific type of pot to begin with. Preferably it is large enough for the tree to grow, but shallow.
However, most pre-bonsai are grown in full ground.
Bonsai are not created by growing young trees into bonsai, but by reducing large trees.
Do understand that it will take years before you have anything worthwhile to work with.
Do stray away from malls, utility stores, roadside stalls and other places that sell mallsai or shops that sell bonsai-seed.