"He can read". What is the verb?

Your example sentence has three words in it, and two of them are verbs (and the 3rd word is a pronoun).

Within the set of verbs, there is a large number of them that are could be labeled as lexical verbs, and there is a very small number of them that could be labeled as auxiliary verbs. There are some properties that differ between the two sets (lexical versus auxiliary) of verbs, e.g. subject-auxiliary inversion which is a property of auxiliary verbs but not of lexical verbs (in general, or almost all the time, this is true). -- caveat: This was a simplified and overgeneralized explanation.

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I've always though verbs are only words like reading, climb, and common stuff like that until I recently learn that words like is, can or be are also verbs. Is there a different name for these kinds of verbs?

There are significant differences between auxiliary verbs and lexical verbs as to the number of inflectional forms that they usually have (but note that the auxiliary verb lexeme BE is quite unusual w.r.t. to this).

If you want more info on this stuff, you'll probably be best served by studying a book on English grammar that's written by a professional linguist.