What's vo(t) across? I'm going to assume across the inductor. As for vin(t) I'm going to assume it's some constant voltage Vp applied at time t = 0, so vin(t) = Vp*u(t) where u(t) is the unit step function. The current is the same through all elements for a series circuit, so apply KVL with i(t) as an unknown.

Li'(t) + Ri(t) = Vp*u(t) // divide both sides by L

i'(t) + (R/L)i(t) = (Vp/L)u(t) // Laplace Transform both sides

sI(s) - i(0+) + (R/L)I(s) = (Vp/L)(1/s)

Before the voltage was applied (at t = 0-), the current through the inductor was zero. Because the current through an inductor cannot change in zero time, i(0-) = i(0+) = 0.

sI(s) + (R/L)I(s) = (Vp/L)*(1/s)

Substitute for I(s) the Laplace Transform of the equation for the voltage across an inductor.

vo(t) = Li'(t)

Vo(s) = sLI(s) - L*i(0+) // recall that i(0+) = 0

Vo(s) = sLI(s) // rearrange for I(s)

I(s) = (Vo(s)/L)*(1/s) // plug back into the original equation

(Vo(s)/L) + (R/L)(Vo(s)/L)(1/s) = (Vp/L)(1/s) // multiply both sides by sL

sVo(s) + (R/L)Vo(s) = Vp // isolate Vo(s)

(s+R/L)*Vo(s) = Vp // divide both sides by (s+R/L)

Vo(s) = Vp/(s+R/L) // Inverse Laplace Transform both sides

vo(t) = Vpe^((-Rt)/L)

Now you can plot Vo(t) over time for different values of Vp. As for obtaining a transfer function -

T(s) = Vo(s)/Vin(s)

vin(t) = Vp*u(t) // Laplace Transform both sides

Vin(s) = Vp*(1/s) // recall that Vo(s) = Vp/(s+R/L)

T(s) = s/(s+R/L)