I am sure that Andrew Wiles had wanted to work on Fermat's Last Theorem even as a graduate student, but surely John Coates advised him not to and instead wait until he has tenure somewhere to pursue this daunting task.

The ideal first research area of a graduate student (in my experience) is one where there is sufficient interest in the area, with a diverse selection of problems (so that it is extremely unlikely to be 'mined out' in the near future), and that there are problems your supervisor know of that can be fairly reasonably attacked by modifying a known method. Preferably after you have geared to solve your first problem the tools you have developed can easily be repurposed to solve a plethora of other problems so that you establish a reputation immediately.

So in summary, dip your feet into the pool first and get acquainted with the broader picture of number theory. If at some point your research naturally crosses paths with transcendence theory, pursue it then.