How does Fourier series apply to signals?
Fourier series can only be used to represent repetitive signals. So if you want to use Fourier series to represent a "signal that should transmit bits", it will have to be a signal that transmits the same bits over and over.


  what do an and bn represent? 


They represent the relative magnitude of the in-phase and quadrature components of the harmonics in your signal.

Which doesn't really tell you anything new. 

What you've really done by taking the Fourier series is found a new way to represent all the information in your signal. Mathematically, you've transformed it to a new basis set.
This is useful because, for example, if you were to pass the signal through a filter with a known frequency response, it would be much easier to calculate the output by using the new frequency domain basis set, than directly using the time-domain representation.


  How do I calculate them?


Your 2nd, 3rd, and 4th equations are exactly how you calculate them.

Two key points. First, c is not a complex number, it is a real, as shown by the 4th equation.

Second, your first equation should be more like

\$g(t) = \dfrac{1}{2}c + \Sigma_{n=1}^{\infty}a_n{}\sin(2\pi{}nf_0t) + ...\$

Note the added n in the argument of the sine, as mentioned in the comments.

Also, notice I use f0 instead of just f. Here f0 is the frequency at which your signal repeats. That is, f0 is \$\dfrac{1}{NT_b}\$, where N is the number of bits in your repeating sequence, and Tb is the period of a single bit.