“I teach myself”

Solution 1:

It is formally correct grammar, following only accepted rules of standard English. However it is not idiomatic; the circumstances would have to be very particular to say that.

The more idiomatic/natural way of saying what you mean by 'I teach myself' is:

I am teaching myself X.

The present simple 'I teach' is just not that common as is in English, whereas the present progressive is so much more common.

As to the choice of 'to learn' or 'to teach oneself' (not the issue of grammar but the issue of lexical meaning), you could just as well say 'I am learning' in English also. But the two are synonymous.

That said, just for fun I checked Google NGrams to see the relative frequency of the three, "am learning", "am teaching myself", and "I teach myself". "Am learning is by far more popular, but it is interesting that "I teach myself" is more common than "am teaching myself". I think this is because, after looking at the examples, "I teach myself" appears (very naturally) in many non-declarative sentences, such as "How do I teach myself", which is a very different context and so sort of doesn't count.

What this means to me is that "I teach myself" or the more natural "I am teaching myself" might be used, but more likely just say it like in Italian "I am learning".