1. what is the smallest tank i would have to have? i know it depends on the size of the turtle, but...
This is the wrong question to ask. Turtles actually do better the more room you give them. The best housing is outdoor pens or ponds with loads of room for them to wander and get their exercise in. Indoors, for aquatic turtles, you will need at the barest minimum 10 gallons of water per inch of turtle (shell measurement) to get good water quality and have space for exercise. If you plan on buying a young turtle, you should research the adult size of the species to know what size home to buy for it. They generally grow very quickly at first in captivity, slowing down once the reach a bit over half their adult size. The common "red eared slider" is a cute two inch turtle for a few dollars in the beginning, but if female can come close to a foot in size, requiring 120 gallons of water.
And the water volume is only part of the size tank you need. If you get a ten inch turtle and therefor need 100 gallons of water at minimum, you have to remember that you cannot fill the tank all the way up, but need to lower the water level enough to put in a basking spot. This means buying a tank big enough to hold 100 gallons of water and several inches of air on top of that.
Indoors, turtles are extremely expensive to initially set up with filters, tanks, lighting, etc. After the very large initial investment, they are not very expensive to maintain (food is cheap, occasional vet visits are usually not too expensive and rare if you put your setup together properly in the beginning). Outdoors, things can be much cheaper, since building materials for pens outside can be less expensive, and you can skip the lighting and such.
2. what are the smallest types of turtles, and approx. price
mud and musk turtles, aquatic, from $10-$20
spotted turtles, aquatic (but also need a land area of some size), from $125-$175
Both of these rarely exceed five inches. A mud turtle could do well in a 60 or larger gallon tank, with a suitable basking spot. The spotted turtle should have a large land area to roam as well as a large shallow water area full of plants.
and last (for now)
3. what id the best things to feed them, or what do they eat?
This depends on the species. Of the smaller varieties, a diet with a suitable turtle pellet (nasco turtle brittle, reptomin, etc) and supplimented with fresh fruits and vegetables and some live foods (earthworms, snails, crickets, all kinds of feeder insects) is good for most species.
The most successful turtle keeper will be one who makes a significant investment to start, and is prepared for the adult size of the animal, and the fact that it may outlive them. Good filtration and a good knowledge of aquarium maintenance is very helpful. Not rushing in, and taking the time to purchase and set up a tank and circulate it for several weeks will assure better results. Taking a new turtle to the vet and having it checked up on is an excellent advantage. Some species may seem cheap to buy, so it may seem odd to go to such trouble and expense to house them, but these initial costs (though pretty large) are important and only need to be paid once.