By "using the definition" they mean

$\large F(s) = \mathscr{L}\left \{f(t) \right \} = \displaystyle{\int_{-\infty}^{\infty}}f(t)e^{-st}~dt$

and for 2(a) you would use "integration by parts" to evaluate this integral.

for 2(b) just use their hint and do the integration.

I should note that it looks like they implicitly mean f(t)=0 for t<0, otherwise these transforms won't converge.

for 4) you should have read about using Laplace transforms to solve linear constant coefficient differential equations. This is a pretty straightforward example assuming H(t) stands for the Heaviside step function. Go re-read that section of your text. Or look at this.