Depends on who you mean as "we". The speed of light isn't a speed limit. If you can create a ship that is capable of 1g acceleration, it doesn't just stop accelerating as it reaches the speed of light relative to some stationary object, like Earth. Instead you start getting relativistic effects and things start getting very weird with time and distance doing some funky stuff. You keep zooming along just fine from your perspective, but an at-rest observer on Earth would see your ship asymptotically approach the speed of light, but never exceed it. The universe is very weird. In any case you could viably travel billions of light years in a single human lifetime, but for an observer at rest billions of their years would genuinely pass. In other words, traveling into the future is very much a real thing, so far as our current understanding of the universe goes.

The search term on this is 'relativistic starship.' Here's [1] a calculator to see what the math works out to for a ship capable of accelerating at 1g indefinitely. So for instance you could travel to Andromeda, some 2 million light years away, in about 28 years. But 2 million years would really pass for those at relative rest, such as those on Earth. So if you came back, the humanity you found (if any) would be unimaginably different.

And this isn't some just some weird fringe theoretical/mathematical thing. For instance GPS satellites have to compensate for time dilation because relativistic effects would otherwise have a substantial effect. Another example is at things like the large hadron collider. As a convenient effect of relativistic effects, emergent unstable particles exist far longer than they 'normally' would before decaying due to the fact they're moving at relativistic rates.

[1] - http://www.convertalot.com/relativistic_star_ship_calculator...