As a singing coach with an additional degree in audio engineering, I have learned ProTools, Logic Pro X & Ableton and although each has amazingly different advantages, I have a favorite.
Logic Pro X is ideal is you have a Mac. In addition to the affordable price of $199, it works well with operating software since it I made by Apple. For singers, many have tried Garage Band and the interface is the same except you will have far more capabilites for production, recording and editing if you shell out the extra cash for Logic. I also find that it's the easiest for recording vocals - specifically multiple takes.
If you have a computer that doesn't work with Logic, my next choice would be Ableton. I find that it is a different kind of learning curve, but it's is FUN. I also find that Ableton is ideal for performing live with tracks.
If you're interested in learning these DAWs (Digital Audio Workstation), feel free to come to our studio for recording or lessons in Venice. We are located right on the border so nestled between Mar Vista and Santa Monica as well. As someone who began on piano and singing, I have lots of tips to help you get started with this software and apply to you needs.