LibGDX is indeed good, this was a question me and my partner in crime asked 8 months ago, make an engine, or go with libGDX.
We went the route of making an engine. After 24,000 lines of code later we are somewhat ready to makes games. It took us monday to friday, programming 9 - 5:30 for past 8 months to get to this point. We have an editor, our own scripting language (incomplete) and an Engine that can port to pc (incomplete) via LWJGL and Android (complete). And unfortunately after all this hard work, we don't even come close to the features that libGDX provides.
Making an engine is not easy, it will require dedication, skill and utmost motivation. I recommend is go with libGDX or equivalent and don't waste time, start making games.
If you're still keen on making an engine i can recommend a few good reads.
Beginning Android Games - We started with this early on, an excellent book, based on early libGDX design.
Game Engine Architecture - Again, excellent read, it will pummel your mind on various ways to approach engine design.
Ericson Real-Time Collision Detection - If you're making an engine, you will need good collision detection that's both accurate and fast, this book is a very good read. Its examples are written in C though.
Programming Game AI by example - Very beginner friendly book, also a very good read for AI.
Game Physics Engine Development - If you plan to have physics and are not very good at math (like me) this book is also a very good read. The engine it develops is not optimal but it does give very good examples.
I hope I didn't scare anyone off. Making an Engine can teach alot. This past 8 months i've learned more developing this software than I did in 4 years of University doing Computer Science.
My experience writing an engine for my game has been pretty similar. I've been working away at it in my spare time for about a year and half now. After the last month of polishing I'm mostly at a point where I can start working on the actual content of my game.
I'm glad went the route that I did but then I like interesting challenges and the engines I initially investigated couldn't do what I wanted them to do.
I didn't learn nearly as much as my degree though its knowledge sure came in helpful. The stuff I learned in the last 18 months while writing the engine was more orthogonal (applied) in nature.