Resistance training is a great way to increase strength, muscle mass, bone density and improve performance. The equipment needed depends on your goals and means.
If you want to get fit, you can use body weight to meet most of your goals while using various areas in your house. An example would be to use a step to put your feet on to perform higher level pushups. You can use a single piece of equipment like a suspension trainer that hangs from your doorway to increase the challenge and more variety. If you have more money, an Olympic weight set, adjustable dumbbells and an adjustable bench will meet most of your needs.
If you want to use weight, but have a limited budget or space you can get a kettlebell. A single kettlebell will allow you to work out your whole body while being cost effective and portable. Bands are another option. Resistance bands are essentially large rubber bands that vary in strengths. These are also cost effective and portable while allowing you to perform a total body workout. If you want more equipment, you can build up all of these items as you progress and have a nice home gym.
If this is not an option, you can join a gym which will have most of this equipment. The downside to a gym is that the money you pay will not yield assets such as gym equipment. Gyms are good if you have limited room and need assistance with exercise.
Choosing the right equipment starts with having focused goals for the outcome you want to achieve physically and monetarily.