For an LTI system, output $y(t)$ is given by
$$y(t) = h(t)\otimes x(t)$$
Where $x(t)$ is input and $h(t)$ is impulse response of the system. The operator $\otimes$ represents convolution.
Convolution operation is mapped into multiplication in Laplace domain. ie,
$$Y(s) = H(s)\times X(s)$$
Where, $Y(s)$, $H(s)$ and $X(s)$ are the Laplace transform of $y(t)$, $h(t)$ and $x(t)$ respectively.
You can use any one of the above equation to solve your problem.
- Find $h(t)$ from $H(s)$ and use first equation. OR
- Find $X(s)$ from $x(t)$, use second equation to find $Y(s)$ and then find $y(t)$ from it.
I prefer second method.