Infinity. The one word that causes shudders to anyone trying to work with it. However, the concept of infinity is not particularly hard to grasp, and therefore, visualizing infinity should not be too much of an issue - provided that a person can learn to be imaginative.
Infinity is simply a word defining something that goes on and on, forever. It is a scientific invention, meant to work with scenarios where something goes beyond fathomable limits. In other words, infinity never ends.
It helps to simply think about infinity in the following manner: if a number increases, and keeps on increasing without ever stopping, the number is going to infinity. This simple sentence is the key to visualizing infinity - and many of the best mathematicians and physicists do the same.
For example, when one says that there are infinitely many prime numbers, it simply means that you can find bigger and bigger prime numbers, and keep on going forever without reaching a number that you can claim as the 'last' prime number.
As another example, one can say there are an infinite number of intervals between 1:00 pm and 1:01 pm. You can have 1:00:01 and 1:00:02, or 1:00:001 and 1:00:002, or 1:00:0001 and 1:00:0002, or so on. You can keep getting smaller, and smaller, but never reach the end. That is the meaning of infinity.
Actually doing the visualization in not something that can be taught. One must learn to do it through practice and imagination. It helps to think of numbers, as said earlier. It also helps to think of the particular case you may be working with.
The real problem is that infinity is rarely truly met outside the world of mathematics, and in mathematics, there are standard ways of dealing with it that do not require any significant visualization. Most 'infinities' are 'approximate': the subject in question does not go on forever. But the quantity / number involved is so large, we call it an infinity. This is called 'tending to infinity' or 'limiting to infinity'. Indeed, this is often what the infinity in physics and astronomy implies. However, these can be thought of in the same way as a real infinity - something very large, or very far: so much so that no matter how we try, we can never reach.
At the end, it comes down to understanding the true meaning of infinity - and learning not to fear it. Many people master infinity, and even enjoy working with it. All it takes is that dash of imagination and belief.