If one is travelling at a significant fraction of $c$, will the length of the trip be shortened? Assuming two stars are 1 light year apart and a traveler is travelling at 0.75 of $c$, from the point of view of the traveler what would be the observed time en route? Also, if a vehicle is constantly accelerating, will it reach 0.75 of $c$ within a reasonable amount of time? What would the Lorenz transforms look like for these situations?
Be gentle, I'm not a physicist. I'm writing book, and I'm trying to devise a scenario whereby a vehicle with a practically unlimited amount of energy can travel between stars. I don't care about the actual travel time only that experienced by the travelers. 
 A: Suppose to have an observer at rest with respect to the star and assume that their distance remains constant (1 light year). A spaceship traveling at $0.75 \, c$ with respect to this observer will cover 1 light year in $1.33$ years (16 months). However, for an observer inside the spaceship the length is contracted by a factor
$$
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1.51
$$
and so the distance of the two stars for this observer is 
$$
d = \frac{1 \;light  \; year}{\gamma} = 0.66 \; light \; years.
$$
The spaceship travels at $0.75 \, c$, so the travel, from the point of view of the observer inside the spaceship, will take
$$
t = \frac{0.66 \; light \; years}{0.75 \, c} = 0.88 \; years \approx 10 \; months.
$$
Of course, going faster than $0.75 \, c$ will reduce as much as you want the time felt by the travelling observer to go from one star to another.
A: The time measured on board the ship during such a trip would be 0.88 years (as calculated by Alex A). If you factor in constant acceleration during the whole trip (starting from zero and passing by the target at maximum speed), the time on board would be 1.9 years and you would be passing by the target at 0.75 c (coincidently), if you accelerate by 1 G. Other acceleration values will result in different times.
You can play with the parameters in various relativistic online calculators:
This one allows for two-way trip calculation
This one involves acceleration during the trip
