Both ways may be correct depending on the context. "Я" means "I" (Nominative case), "Мне" means "for me" (Dative case).

See case system of Russian language in any textbook.

Examples:

- Я хочу яблоко.
- Я тоже.

- I want an apple.
- Me too.

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- Дай мне яблоко.
- Мне тоже.

- Give me an apple.
- Me too.