I think there are two questions involved here. One is metaphysical: "What does time mean to me?" or a similar thought. Can't help you with that one. The other: "Is time real?" is more interesting from my point of view.
Time is relative: it happens here, to me, differently than it happens there, to you. How fast time passes here depends on my position in relation to a gravity well and my relative motion. The same goes for you - but relative to your location and motion. But one thing is sure: we are both heading in the same direction. We are both getting older. Now, if I'm on a global positioning satellite geostationary above you, I am travelling much faster thn you are and there is a corresponding difference in how fast time is passing. I can observe you getting older more quickly than I am: you can observe me aging more slowly. If, however, we both meet up at either your place or mine, we will then be aging at the same rate.
So, is time real? I think the answer is that time is as real as space, so we could be heading into semantics territory here. Physicists combine space and time into a four-dimensional "spacetime" which inextricably relates the measurement of space and the measurement of time together and neatly explains why speed (through space) affects the passage of time. Logically, one could not have space without time and vice versa, so the question "Is time real?" should be phrased "Is spacetime real?", to which the answer is "yes".
Outside sci-fi, time travel as such remains impossible. I can speed up or slow down what Prof Hawkings calls "The Arrow of Time" by changing my relative motion through space. If I travel at speeds approaching the speed of light, you will observe me aging very slowly; but I will observe myself aging at the same rate as I always have.