This is my first time using DigitalOcean. Considering this and that I'm planning to serve a Laravel application, should I use a LAMP Stack or PuPHPet configured server?
It's hard to say one way or another without knowing more of your requirements. The DigitalOcean LAMP One-Click app give you a server with Apache, MySql, and PHP pre-installed. It's quite likely that you will need to do some additional configuration, depending on what you are looking to accomplish but it should give you a solid base to build on. The PuPHPet wizard allows you to pre-configure a number of additional bits. It might be a good option if you are less familiar with the Linux command line.
Love to hear about other users experiences with PuPHPet.