Step one, have the app on a separate subdomain from your marketing site.
Initial version, use Unbounce or Squarespace or whatever gets you moving with the least wasted effort.
Later on, you probably want a CMS so it's easy for non-technical users to update. WordPress is happy because of adoption levels, then throw money at WPEngine to handle many of the more common issues for you.
You could use Drupal or some custom thing if you WANT...but make sure you have a benefit equation that justifies it. The more custom your solution is, the more maintenance overhead you commit yourself to.
For stuff like landing pages, set up a site style guide. It may also be helpful to standardize on a style library like Bootstrap or Bourbon Neat. Don't make people reinvent the wheel every time.
Then add tools like Optimizely, Woopra, Kissmetrics, Intercom, Snappy. Your major bases to cover are generally testing, tracking, and customer interaction.
It may also be helpful to add a content management tool like GatherContent. WordPress may be easy to update but planning what to update and aggregating assets for it is a completely different problem.