How do i go about finding the transfer function of this system with two inputs?
Considering your block diagram with the signs showed as in the figure below:



The principle of superposition can be applied in this linear system in order to investigate the effects that two inputs (regulation \$r(t)\$ and disturbance \$d(t)\$) on output. In this way, we have the two transfer functions in \$s\$ domain:

Doing \$d(t) = 0\$:
$$ G_1(s) = \frac{Y(s)}{R(s)}$$
and doing \$r(t)=0\$:
$$ G_2(s) = \frac{Y(s)}{D(s)} $$:

Then, using the standard feedback formula:
$$ G_1(s) = \frac{K}{s^2+3s+2+K} $$
With \$K=10\$:
$$ G_1(s) = \frac{10}{s^2+3s+12} $$

For \$G_2(s)\$ determination, note the negative sign on input of block \$K\$:
$$ Y(s) = -\frac{K}{(s+1)(s+2)}Y(s) + D(s) $$
Or:
$$ G_2(s) = \frac{s^2+3s+2}{s^2+3s+2+K} $$

With \$K=10\$:
$$ G_2(s) = \frac{s^2+3s+2}{s^2+3s+12} $$
The expression for output is:
$$ Y(s) = G_1(s)R(s) + G_2(s)D(s) $$

Finally, replace both \$R(s)\$ and \$D(s)\$ by \$1/s\$ (the Laplace transform of unit step input \$u(t)\$).

$$Y(s) = \frac{1}{s}$$
An simulation graph: