I believe, if you set up a repo from scratch in Coda you need to use the terminal to add the remote path. Open a Terminal tab and paste in whatever Beanstalk is telling you for 'Adding an existing repo' or something along those lines. It will be something like: $ git remote add origin git@<username>.beanstalkapp.com:/<repo>.git
. After you do that, to Push to that repo you can use the Push button at the bottom of the Source tab.
As far as processes go, what you're doing with Beanstalk is probably the "simplest", that is, it is probably the one with the least amount of time required on the command line or installing software on your server. I would say the next step would be deploying straight to the server by setting it up as a git remote. You could then SSH in to the server and pull changes directly. After that you have tools like Fabric and Capistrano which automate the process.