We will begin with some basic stuff about Dragonball Z. The first thing is that it is a work of fiction and fantasy. You should never do ANYTHING that is shown on the program, nor should you replicate their techniques. You will probably die if you try. Luckily there are no artificial gravity machines to flatten you like a pancake.
When finding ways to train like the Z fighters, you should remember the most obvious. Pushups, Situps and Running. You see Goku doing all of those. Now, he can do them in excess of 100x earth's normal gravity, but you can do it the non-saiyan way. Here is how:
#1 Pushups- There are a nearly infinite amount of ways to do pushups, but my favorite are knuckle pushups. They toughen your fists and lock your wrists in to place making your back muscles, triceps, and chest work. Try putting your feet at a six-inch elevation and wearing a weight vest. This will be a little closer to 1.5x earth's gravity. Still better than nothing.
#2 Situps- The only way to have powerful kicks is to increase your core strength. Power comes from the hips, balance is aided by the obliques, and the quads push you through your target. The best way to do situps is on a decline surface. You can do this by performing your situps on a hill with your head lower than your butt. It will double the resistance if you get a good enough incline, increasing your training potential to 2x earth's gravity.
#3 Running- Please use caution while running. It is far and wide the best solo cardio you can perform, but it can also destroy your knees. The more weight you carry, the more knee damage you will endure. To help alleviate the stress to your most presious joint, run by striking the ball of the foot instead of the heel. It creates less of a shock going through your body and builds your calf muscles faster. In turn, your punching power will increase greatly, because you will be able to give the ground more impact by pushing harder with a strong calf muscle and an uninjured knee joint. This is kinteic linking at its finest.
#4 Sparring- It is the one thing you see every fighter doing on DBZ. You get better at fighting by fighting. Please use all available protection and consult a doctor before fighting or any other physical activity.