Since no one has seen fit to include the obvious..

A likely first time machine, and perhaps the only possible variety, will look more like a television than a car or a telephone booth.

It is much more practical to view the past and the future than to visit. Any correct theory of the Universe, including the Big Bang, that describes a beginning with definable parameters allows the creation of such a time machine.

If you can determine the state of all matter in the Universe at any one moment (such as the beginning), and you understand how all matter interacts (gravity, fusion, lepton decay, etc.), then you can input this information into a supercomputer, press the Play button, and watch the Universe progress from the very start.

You would, naturally, be able to jump around from any time to any other time. Watch the dinosaurs go extinct, see who killed JFK, and what happened to those Anasazi again?

A big plus to a viewing time machine is the lack of paradoxes. The machine is naturally self-aware and aware of the viewer: their matter is subject to the same interactions as all other matter. The past is not influenced at all by the viewing. The future that the machine shows you is unchangeable-- that you used the machine to view the future, and any actions you took as a result of that, have already been included in the output you're viewing.

If while using the machine you had seen your best friend killed in a car accident, for example, and told them not to drive that day, the machine would show you watching it, seeing your friend die, telling your friend not to drive, and your friend surviving. There is no paradox there at all.

A simpler and more accurate example, however, would be a vase sitting next to the machine. In 10 seconds a car is going to backfire outside, causing the vase to fall and break. You are using the machine to view yourself 10 seconds in the future, so you see the vase fall and decide to catch it as soon as you hear the car backfire. The catch here is that you didn't just see the vase fall, you see yourself watching the vase fall, deciding to catch the vase, and catching the vase. You can take this to tens of levels of depth in your mind and on the monitor (like looking into a mirror with another mirror facing it on the opposite wall), but at the end of the 10 seconds you will have made a decision whether to save the vase or not-- and this is what the machine will show. You can see then, how events 10 years in the future cannot be changed either; all of your actions from now until that point 10 years in the future have already been accounted for.

So you can, in fact, alter the future that would have happened if you hadn't used the time machine, but you cannot alter the future the time machine shows you. Complex, yes. Paradoxical, no.

How, exactly, can a supercomputer, composed of a tiny percentage of the total number of particles in the Universe, keep track of all of the particles (including photons, gravitrons, etc.)? The principles for such a feat are already in use in today's programming: shortcuts, workarounds, and dirty hacks. You could, for example, mark off entire cubed light years as "vacuum" or entire cubed meters as "hydrogen". You could treat most atoms as indivisible most of the time, and only track subatomic particles when needed. While there would undoubtedly need to be thousands or hundreds of thousands of such hacks, it is definitely within reason to assume such a machine could be built.

And of course, with the right VR equipment and enough computing power, you could even play along. Let's get started.