Depending on how far you wish to go, you can raise them in specially-made butterfly cages or make your own out of say, a 5 gal. paint bucket you can purchase at one of the big box do-it-yourself home improvement centers.

As for a butterfly cage, I made one out of a collapsible clothes hamper. I bought this particular one on Amazon. Considerably cheaper than buying a collapsible butterfly cage, which many are just re-branded clothes hampers. In green. More expensive. Yada-yada...

I also use a "cat bucket" at times made out of a 5 gal. paint bucket. Cutting a hole in the top and screening it like the photo below allows air circulation and viewing. A couple of branches inside help the cats to climb towards the top in order to pupate.

Place whatever type of container in a shaded area outside so the caterpillars don't fry in the sun.

Raising Butterflies has some good info on raising cats.

An alternative to bringing your cats into a cage is to cover the branch of the host plant with a mesh sleeve. Once you have eggs or cats on a plant, you can enclose that branch of the plant in a sleeve similar to this:

http://www.watchbugsgrow.com/caterpillar-habitats/sleeve-habitat.html

A less expensive version would be to use a one gallon or five gallon mesh paint strainer from the hardware store or go to a fabric store and purchase some very fine mesh bridal veil fabric called "tulle" and wrap the branch in that. Once you have have the eggs/caterpillars enclosed, they can mature relatively safely on the plant. You will need to open the sleeve and shake out the frass as it accumulates.

I thought the cats moved to a different plant. If I enclosed them they couldn't do that?

Keep an eye on the amount of leaves left inside the sleeve, to insure there is enough for the caterpillar to finish growing. Remember that they eat a ton during the last two instars. If the leaves are running out, transfer the caterpillar to another plant and cover that branch with the sleeve. Or, you could let nature do what it will. Keep planting milkweeds to give as many Monarchs as possible a chance. The tachinid flies need to live, too.