For Monterrey Jack, you will need:
Milk - fresh raw is best, but non-homogenized pasteurized or homogenized pasteurized will work so long as it is not ultra-pasteurized (high heat pasteurized)
Mesophilic culture - you buy this in small packets suitable for 2 to 4 gallons of milk, or you can make your own. There are several kinds of mesophilic cultures that produce good cheese. To make some, buy a quart of the freshest cultured buttermilk you can find and leave it on the counter at room temp overnight, about 6 to 8 hours after it reaches room temp. It'll thicken to a yogurt-like consistency. To innoculate your milk, you'll need about 4 to 5 ounces per gallon of milk. You can freeze whatever you don't use immediately it in a sanitized ice tray, then put it in a baggie and store it in your freezer.
Calcium Chloride. This is usually sold in a 30% solution, but it can vary. It will usually say how much per gallon of pasteurized or homogenized milk. Not needed if you are using fresh raw milk
Rennet - you can buy tablets or liquid. Amount needed per gallon depends on the type and is usually specified on the package. Typically, for vegetable rennet tablets, use 1/4 tablet for 2 gallons of milk. Typically, for single strength animal rennet, 1/4 tsp per gallon of milk. The amount needed varies by the type of cheese you are making also, so check the cheese recipe.
An 18% to 22% saturated brine, usually specified by the recipe. Use non-iodized salt. You can google it, but a common brine is around 21%, 2.25 pounds of salt per US gallon of water. Add 1 teaspoon vinegar, and 1 tablespoon calcium chloride solution for one gallon of brine. The brine can be kept in the fridge and reused.
A mold - a 6 inch diameter mold about 6 to 8 inches tall works well for 2 to 4 gallons of milk. You can improvise a mold. I've seen sites on the internet where people used a #10 tin can. You need a way for whey to drain, so usually a mold for a hard (or semi-hard) cheese like Monterrey Jack will be open on the bottom.
A press is nice to have, but you can simply put a follower of the same size as your mold on top of the curds and use weights, jars of water, books, etc, to press it down. The recipe you use will generally say how much weight is needed.
Thermometer to measure milk temperature
Bamboo or plastic mat to dry the cheese on.
Some butter muslin and a collander to drain the curds.
A fine weave cheese cloth to line the mold.
There are kits you can buy that include the basic equipment and ingredients, minus milk and major items like a press.
I recommend you get a good recipe book, or hunt down a recipe on the net. Cheesemaking books usually have a great deal of information about the processes involved in home cheesemaking, so I think they are worth buying.