Those sentences aren't wrong grammatically, but they both sound unnatural.
The most natural way to say this is 'I've started studying French again.'
25. November 2015
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Both are correct. #2 is more formal and might be used in writing.
English speakers will assume that words like "French" or "Italian" refer to the language unless something follows, such as in "French architecture" or "Italian food" etc.
25. November 2015
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