What you "have to learn" to be able to use some of the new app builders is minimal, especially if you use the AGOL version (and understand GIS)...however, you're on the right track on what is good to know. JavaScript (and Html5/css) will help you quite a bit in understanding how to implement the custom widgets available if you get into using something like the Web AppBuilder developer's edition (WAB) To find out more about all that, I recommend looking at my blog Web AppBuilder Developer Edition - Customization Resource List Which has links to all types of info.
WAB is for web applications, not native iOS or android, but because it uses JavaScript it will work on all mobile devices (unlike Flex and Silverlight). For native apps, you should check out AppStudio for ArcGIS | ArcGIS (still in beta) which is meant to be "build once - deploy to many" and will create native applications that are ready for the stores without having to learn the details of the OS development.
Python is create to know for many uses when it comes to GIS, although not directly for use with mobile applications. When Python, and python scripts will come in handy is perpping data for you apps, and creating geoprocessing tasks.
One thing that you should get comfortable with are IDE (programming interfaces what help with syntax, etc) for each of the languages. For Python, I personally really like Download Wing IDE Pro v. 5.1.7 - Wingware Python IDE I use the "Pro" version, but here is a free version available too.
For Javascript, I use both Brackets - A modern, open source code editor that understands web design. and Notepad++ Home both are free and have plugins that can help tremendously with syntax and formatting. Sublime Text: The text editor you'll fall in love with is another popular one.
What IDE is best is so subjective and really depends on what you get used to. And sometimes the best way to learn any of this is to come up with a project and just jump in to see what you can do. There doesn't seem to be one complete solution for all (at least not right out of the box).....it really depends on what you need to do. So have fun, keep learning, and take advantage of GeoNet and all the free Esri Video | Videos Covering GIS Events, Products, People, & Topics and free or cheap training at Esri Training
