I think that depends. If you're doing it as a learning exercise then it can be beneficial to write all the code yourself, at least so you can appreciate the stuff that a framework usually takes care of for you, and to have some idea of how it works behind the scenes (to this end, it's also good to read through the code of some different frameworks to see how they approach common tasks).
On the other hand, once you're past the learning curve of a particular framework you can be a lot more productive, as you avoid writing a lot of 'boilerplate' code for every project. I'd also definitely look into a few of the most popular ones if you want to improve your job options as a web developer (Symfony2, ZF2, and Laravel being some of the most popular).
As for using Laravel for a project like the one you're doing, there are some pros and cons. On the plus side, Laravel seems really nice to work with (I've just started playing around with it myself) and is certainly very popular, so it's easy to find tutorials and bundles (modules) for it. It also has some nice features that make it easy to put together a RESTful API. On the con side, the codebase is quite large (roughly 18mb for the base install).. it includes a lot of stuff that might be overkill for a simple API that's serving as the back-end for a JS app. If you were building an API that was going to get heavy use, I suspect Slim might give you better performance from being rather lightweight in comparison.