You have written:
I've been told ArcObjects and .NET is a thing of the past and to focus
on Java, is this true?
.NET is definitely not a thing of the past. Microsoft continues developing this platform and this is by far one of the most popular one to develop Windows based applications and services. Learning .NET plus C# is definitely good to be attractive as a GIS Developer but also as a Developer per se. You can do heavy customization of most large commercial GIS desktop and server products (MapInfo, Autodesk, Esri). You will also be able to write own GIS applications and services as well as administer GIS services and automate a lot of work.
Java is very popular indeed because of no license hustle (it is free) and because it is cross-platform and is installed on most devices. Mastering Java imho takes longer time comparing to .NET/C# but I am not a developer originally and was exposed more to Microsoft stack. Java skills won't hurt you, but for most job ads I've seen it does tend to be more .NET rather than Java. However, as blah238 mentioned, for the open source software Java would be slightly more popular.
Since you don't have worked with either .NET or Java, I'd recommend sticking to .NET rather than Java if you are going to spend most of your time working with commercial GIS software rather than open source GIS and if you aren't going to write own GIS systems for mobile devices or Linux.
What else are good gis/dev skillsets to have?
I usually refer to the Michalis Avraam blog post whenever people ask which way to go.
Since you already know Python, keep learning it. Python is extremely popular both in commercial software (it's Esri's primary scripting tool for the Desktop platform and new ArcGIS Pro will support Python 3.4 plus strong geoprocessing platform; customization of largest open source GIS such as QGIS is possible with Python, too; GIS services administration and automation etc.). So I would spend most of the time with Python because it is so much worth it. A quick search on Amazon would show how popular Python is in geospatial community; it is also widely used in many other industries and academia and I believe this will remain so for a good time.