It depends. A collection of free weights of graduated weight (or sets that come with magnetized extra weights that attach to barbells) are a good way to do strength training at home. If you are just starting, it might be helpful to work with a trainer once or twice to make sure your form is good and avoid injuries. Always start with low weights and only add weight when it is easy to do 8-12 repetitions.
No one answer. Although there are many ways i recommend functional movement types of strength training. These types of exercise carry over into everyday activities and are the best "bang for your buck" in my opinion. These types of resistance exercises are often done using kettle bells and you can find lots of home workouts on the internet. Good form in paramount to this type of training.