You could translate "I've just recently started learning German" (almost) literally. There a couple of possibilities to translate just recently. A few examples:
Ich habe kürzlichst erst angefangen Deutsch zu lernen.
Ich habe erst kürzlich angefangen Deutsch zu lernen.
Ich habe vor Kurzem erst angefangen Deutsch zu lernen.
The first version is restricted to some regions of Germany though.
Your sentence is not really wrong, though. "Ich fange an", however, is either present or future.
Ich fange gerade an. (I'm starting right now.)
Ich fange morgen an. (I'm starting tomorrow.)
But you probably want to say that you've already started, so you should go with Present Perfect.
It's important to mention that the tenses are very different in German, although these simple examples suggest they're equal.