How can I calculate the amplitudes of this signal's harmonics?
You can find the harmonics of a signal by computing the Fourier series of the signal, given that the signal is periodic. 

You have a non-periodic signal here . However, you can still calculate Fourier series between the full finite interval \$T\$ of the signal, assuming that the signal is 'periodic' with \$T\$, ie., assuming that the signal repeats over and over like that afterwards. 


  How should I derive the Fourier transform on this signal?


You can calculate the Fourier transform of the signal using integration by parts. For eg:


$$F{\{v(t)\}} = \int_{-\infty}^\infty v(t)e^{-j\omega t}dt$$
$$=\int_{0}^T v(t)e^{-j\omega t}dt$$
$$=\int_{0}^{t_0} v(t)e^{-j\omega t}dt+\int_{t_0}^{t_1} v(t)e^{-j\omega t}dt+\int_{t_1}^{t_2} v(t)e^{-j\omega t}dt+\int_{t_2}^{T} v(t)e^{-j\omega t}dt$$

Each of the four terms are simple linear equations for line on the respective intervals, which you can derive if you know the values of \$a, b, c, t0, t1, t2, T \$