Space Travel and length of time

You're actually describing the twin paradox.
That's a little beyond Special Relativity, since the astronaut accelerates, and deaccelerates. Since Special Relativity treats only inertial frames of reference, accurate treatment would require some extension towards General Relativity.
But you may consider two astronauts, one travelling with constant velocity towards Earth, the other one with the same (by amount) velocity away from the Earth, and then apply the formula for time dilation of Special Relativity to both astronauts, as an approximation, and to avoid the twin paradox.
Then you get the almost ubiquitous factor $sqrt{1-beta^2}$ of Special Relativity, with $b=v/c$ the ratio of the relative velocity to the speed of light. For $beta=0.1$, hence 10% the speed of light, we get a factor of

$sqrt{1-beta^2}=sqrt{1-0.1^2}=sqrt{1-0.01}=sqrt{0.99}approx 0.9949874371$.

Divide 10 years by 0.9949874371, and you get 10 years plus 18.4 days on Earth.

For $beta=1$ this division would fail, implying a break-down of the formulas of Special Relativity. That's why travelling with the speed of light doesn't work for astronauts respecting Special Relativity.