I would say that infinity is one of those things that we can only understand through some other medium, like mathematics. Just like we can't see a fourth spatial dimension except by mathematical translation, there are aspects of reality that we never evolved the capacity to perceive with our senses. No matter how hard you try, you can't "see" infinity, because it goes on forever and the means we use to see do not. That does not necessarily mean it doesn't exist.
As far as the universe being infinite, that is difficult to say and your answer will depend on what cosmological theories you subscribe to. Some people imagine that the universe is infinite emptiness, and that the stuff we see is simply flying through that emptiness. But some people have an idea called Dark Energy or Vacuum Energy, which is basically the "cost of having space." According to Quantum Field Theory, every fundamental field must be quantized to every point in space--for example, the gravity of planet Earth exerts a minute but definitely real and quantifiable effect on the entire rest of the universe, and on every point within it. It doesn't affect things on a large scale beyond a certain distance (the old inverse square law), but it affects everything to some degree.
The universe appears to be expanding, and therefore this Dark Energy is thought to exert a negative pressure effect (that is, it pushes everything apart). However, if the universe were infinite, then there would be infinite Dark Energy as well, and the universe would be accelerating apart infinitly fast, which means we would have experienced the Big Rip already. The fact that there is a universe at all would imply that it is finite, that there is a limit to the amount of Dark Energy out there.
For another weird brush with infinity, read about these things called Fractals. You can find images of them on the internet because they are very pretty and interesting to look at. They are infinitely complex mathematical objects--if you graph them, you can zoom in on any section and you will keep finding detail no matter how far you zoom in. Likewise, you can zoom out and you will keep finding bigger and bigger structures that incorporate what you see at first. It is also interesting to note that you tend to see a trait called self-similarity--that is, the fractal seems to repeat its patterns as you zoom in and out, repeating them at all different scales. Many objects in nature exist as fractals--shorelines, sponges, etc. Basically, they are infinity contained within a finite space. A fractal sponge has infinite surface area but finite volume. A fractal line, like a Koch Snowflake, is infinitely long but is a closed loop.
Whatever its true nature is, it is a wonderfully fascinating universe we live in!