How far can we see when we look up?

That is really a question of what there is in space to see. If you're going to be able to see a single object far off, it had better be big and bright. The objects that don't shine by their own light (and we can see really well) extend out to Saturn, which is about a billion miles. If you were to hit the button of a flashlight, it would take the light an hour and a half to go a billion miles. And yet light goes over 7 times the distance around the Earth in a second! Astronomers would call a billion miles "1.5 light-hours." The smallest and farthest planet, Pluto, averages about 5.5 light-hours away.

What is a "light-year."

A light-year is just the extension of the "light-hours" in the question above to larger units. A light-year is the distance light can go in a year. A light-year is NOT a unit of time, it is a unit of distance.

Does that have anything to do with the "stardate" in the old Star Trek episodes?

"Stardate" is just something they made up.

How far are the stars?

The nearest star, of course, is the Sun, and it is only 8 light-minutes away. The nearest nighttime star is a system of three stars always below Panama City's southern horizon. This system's name is Alpha Centauri. It is over 4 light-years away.

What is the farthest star that we can see?

Without telescopic aid, we could probably see a very bright star 10,000 light years away and still perceive it as one star. Such a star actually exists, but it is closer and brighter. We see the star Deneb as one of the 25 brightest stars at 1600 light-years. It would be just visible at 10 times the distance. More commonly at such remote locations we usually see thousands or millions of stars formed in a cloudlike grouping called the Milky Way.

How far could we get away from the Sun and still see it without a telescope.

Our Sun is a star of mediocre brightness. If you were to go about 40 light-years away it would be barely visible, but only if you had good eyes.