Given your level of knowledge and the requirement to power other things from 12V then 12V is probably simplest. An Arduino can run on 12V, but beware that it is not a power supply and can only supply small amounts of current to other devices, such as a small display or some LEDs. Ardunos actually run on 5V or 3V3, in the case of a Uno 5V.
I personally prefer lead acid batteries to lithium because lead acid barbarities are cheap, don't have fussy charging requirements and are resistant to abuse, in particular they are not prone to catching fire.
General advice is start simple, get one thing working to your satisfaction then add a second thing.
You must learn how to write non blocking code, which means learning to use millis, and learning what a state machine is, and never using delay or while loops.
To get you started:
Blink without delay, which is in the IDE examples under 'digital'.