Other than dumbbells, barbells and home gyms, you can utilize specialized equipment to make weight training easier and more effective -- at home or the gym.
Here is a list of suggested ancillary equipment.
If you lift heavy weights, you can buy special shoes designed specifically for the purpose. These tend not to be running shoes or even modified running shoes. The support areas are different and you don't necessarily need the soft cushioning required in certain key areas of a running shoe for repetitive pounding of the ground. Lifting shoes tend to be heavier and with a very solid grip of the lifting surface and a very stable heel. Allow for toe movement but not too much.
Professional weightlifting shoes should not be confused with cross-trainers that are designed for more movement.
Gloves and Straps
Weightlifting gloves and straps give you a nice training aid when you lift heavy for strength training. Use them for deadlifts in particular. The straps wrap around you wrist and provide support. Gloves provide extra grip and support.
Weightlifting belts are used in Olympic weightlifting and in Powerlifting competition. A weightlifting belt reduces stress on the lower back in upright lifts like deadlifts and squats, and prevents back hyperextension during overhead lifts like the military press. Check approved specifications before purchasing a belt if you intend to use it in competition.
Some research has shown that belts inhibit natural abdominal and core strength development.
Hand Grip Exercisers
It's often difficult to find convenient equipment to exercise the forearm muscles. You can do barbell reverse curls and a few other exercises, but having a specific piece of equipment like grip handles around the house can encourage regular exercise. The wrist joint can be a little susceptible to overuse strain, so don't overdo it. Rubber balls are also useful.
Bands and Tubes
Stretch bands and tubes have many uses, and you can pack them for travel as well. Many inventive resistance exercises can be devised using simple stretch bands and tubes. They can be especially useful when you need to incorporate regular injury recovery exercises in daily routine. Keep a range handy.
Fitness and Medicine Balls
Most regular gym trainers know and use the fitness balls that you lie across or sit on to exercise. The heavy medicine balls also go way back as gym tools and you can see that one of their original use was in recuperating from injury or illness. Both devices are especially useful for dynamic, repetitive exercises that target core muscles.
Boxing is not only a good aerobic conditioner, but you get an upper-body and leg workout as well. Learn some boxing skills, get a set of gloves and do some bag work.
The idea of training vests is to add extra weight and thus resistance to any training program. It's one way of increasing training work done with the same routine.
Not really necessary, but some trainers love them for how they target the abdominals. Hard work, but good results.
Jump Rope / Skipping Rope
One of the oldest forms of aerobic training (and playing), 'jump rope,' or 'skipping rope' in other cultures is still popular. But 'ropes' have come a long way since the days of a piece of old rope. New, sophisticated, high-technology ropes give you a new perspective on this form of exercise.
You might not need all of the above, but choosing the right supplementary equipment can add variety and results to you workouts.