Learn Polish: Uczę się nie tylko polskiego, ale także niemieckiego. - I am learning not only Polish but also German.

The reflexive verb uczyć się (to learn) always takes its object in the genitive case. While many verbs use the accusative for direct objects, uczyć się is a strict exception you have to memorize.

Why do polskiego and niemieckiego look like adjectives instead of nouns?

They actually are adjectives! In Polish, when talking about languages, the noun języka (language, which would also be in the genitive case here) is usually dropped because it is obvious from the context. You are literally saying 'I am learning not only the Polish (language), but also the German (language).'

Can I use ale też instead of ale także here?

Yes, absolutely. The construction nie tylko..., ale także... (not only..., but also...) is very common, but you can swap także with też. Także just sounds slightly more formal or emphatic in written Polish.

Could I move się to a different part of the sentence?

In this sentence, right after uczę is the most natural spot. The word się likes to be in the second position of a clause and sounds unnatural if placed at the very beginning or the very end of a sentence.