Congrats on deciding to take control of your health and fitness. If this is your first foray into training, you may wish to consider bodyweight exercises first. There's relatively little cost and your chance of injury is minimal, but, not zero. If you'd still like to purchase a home gym, you should research each machine you're interested in before making a purchase. Knowing that you want to concentrate on back, biceps and triceps, examine the machine(s) to determine how and if they can help you achieve that goal. For example, how many exercise movements can you perform om the machine for a particular body part? Do some research on the types of resistance they use. Some are plate loaded, while others use bands, or, hydraulics. Some people claim they get a better training effect from using plates. While others like bands. If you can, try out the machine before purchasing. Then, measure the space where you will use the machine. Will it fit? Is the space accessible for installing a machine? These are some of the questions to consider before spending your money.
It seems that incorrect exercise movements can cause injury even if a
person has taken care of warming up and stretching before the
training, is this correct?
While machines can provide the advantage of not requiring a “spotter”, don't make the mistake of assuming that they are injury free. Any exercise, if done incorrectly, can injure you.
Are there specific exercises for triceps, biceps, and back strength
that are more prone to injury, if done incorrectly. If so, what are
these and what should a person keep in mind to reduce the risk of
injury in these cases?
As for specific exercises, proper form is essential while exercising any body part regardless of the exercise. I would recommend starting with basic movements like triceps extensions, biceps curls, and lat pull downs for the back. Make sure not to put your body in an uncomfortable position, and, make sure your lower back is supported properly while doing the exercises. More than likely, the machine will come with literature explaining the exercises you can perform. Use that as your starting reference and pay strict attention to any warnings the literature provides. Good luck, and if you have more questions, post them here.