Determine the Zero-State Response and Zero-Input Response
For the zero state: Find

$$
F(s) =\frac{1} {(s-3)}
$$

Which is computed by taking the Laplace transform of course.
Now, multiply F(s) with your transfer function.
You will have
$$
Y(s) = \frac{s} { (s-3)(s^2+s+1)}
$$

Now simply, use partial fraction and take Laplace inverse to find y(t)

Solution on wolframalpha: 

Wolframalpha



As for the zero state response:
You can find the differential equation by doing the cross multiplication
As for s^2 is second derivative and "s" is first derivative:
$$
y' ' + y' + y = f '
$$
Take Laplace transform again considering initial conditions:
$$
s^2Y(s) - sy(0) - y'(0) + sY(s) - y(o) + Y(s) = sF(s) - f(0)
$$
No as for the input F(s) = 0 (for zero input)

So,
$$
Y(s) = \frac{sy(0) + y'(0) +y(0)}{(s^2 +s +1)}
$$
Now sub for the initial conditions and take Laplace inverse to find y(t).