Having a home gym is very important for a lot of people because that allows them to not have to go out and spend money to attend a gym away from their home, as that alone can be a deterrent for a lot of people not to go to the gym. However there is a lot to consider about building a home gym, things like the cost of the equipment, which equipment to buy and what equipment you don't need to buy. 

For starting off at your own home gym I believe the most important purchase would be a large mat that you can put weights on and that would not hurt your body if you had to lay down for some core exercises. This item can often be overlooked initially because its not the item that will directly link in your mind to working out, you will probably think of weights and machines but not the essential starting piece being the mat.

Then you may consider a bench, however this can often be substituted with some other items in the house as long as you can lay on it with room for your arms to move around and as long as you aren't doing any very strenuous workout on it.

Weights is probably the next on the list and probably the most important for actually working out, and when I say weights I mean free weights like dumbbells' or powerbocks as they can be used in hundreds of different ways to benefit almost every workout you could want to do.

The machines may interest you as they are great tools that will simplify your workout and help you perform the workout in proper form, they are a bit advanced tools as they are very heavy and take up a fair amount of space but are great for enhancing our exercise experience and the potential gains you may see.