We have developed lightning App . It uses a lot of Lightning components, Apex classes. We are exploring on how to move our Lightning App to the Bitbucket. Also let me know what is the Best IDE that we can use so that it best supports Lightning.
Apart from the ones mentioned as Atom and Force.com IDE ,using v7 of mavensmate also supports lightning editing
I have been using Dave Carroll plugin for building lightning components for Sublime which uses Force.com CLI under the hood .It gets the aura folder and then I can place the folder in src and have the bundle added in the package.xml file .
The xml for the lightning package bundle in package.xml is as below
I would use the Force.com migration tool to sync all of the metadata to your machine. Then you can do a "git init" and "git commit -am 'initial commit'" to capture everything in your organization. Simply push that repo to the remote for BitBucket.
Force.com IDE will not necessarily sync all metadata component types, though it supports most of them.