When you're first getting started making chain maille, there is the tendency to hold onto your pliers for dear life for fear your rings will go flying across the room. It happens, trust me: been there, done that.
The first thing you have to do is relax.
Find a pair of pliers that are comfortable in your hands and that you feel comfortable with. I highly recommend using a pair with a box joint or spring - this will help you easily and effortlessly work with your tools.
Know that every metal you work with will be different and it might take you some time to get used to each of them. As a beginner, work in bright aluminum or copper. They aren't terribly expensive metals, so if you mark them up as you're getting used to mailling and your pliers, it's not much of a waste. They are also very malleable metals, so you won't have to work as hard to get them closed. Keep in mind that because they are malleable (soft) that means they will mar easily in comparison to, say, stainless steel. This, however, will help teach you just how strong (or light) a grip you really need.
Sit in a comfortable position where you can easily see your work with good lighting and relax your hands. You want your grip only to be firm enough to keep the ring in your pliers. Let your tools do the work - you will exert less energy, not get tired as fast, and be able to work longer.
It will take some practice to get used to how much pressure you really need, so be patient and feel it out. You'll get it.