"I'm Spanish" or "I'm a Spanish"?

Yes, adding an "a" before Spanish, would be wrong. That could be reworded as I'm a Spaniard. See http://dictionary.cambridge.org/dictionary/british/spaniard?q=Spaniard

"Spanish" is an adjective, not a noun. You cannot use either an indefinite article with it. You can say:

• I am Spanish
• I am a Spaniard

However, the two sentences are not equivalent. The first implies descent, while the second implies citizenship or nationality. I could very easily be of English descent, and yet be a Spaniard, if I changed my citizenship, for example.

Spanish is an adjective, so no article. A Spanish man is a Spaniard. Note that for many other nationalities, the form of the adjective and the noun is the same:

• American, an American
• German, a German
• Italian, an Italian
• Russian, a Russian
• Chinese, a Chinese
• Japanese, a Japanese
• Greek, a Greek

I have a feeling that for most nationalities the adjective and the noun have the same form, and only in a few cases are the forms of the adjective and noun distinct, e.g.:

• Danish, a Dane
• English, an Englishman
• French, a Frenchman
• Irish, an Irishman
• Scottish, a Scot(sman)
• Spanish, a Spaniard
• Welsh, a Welshman

Spanish is an adjective. You cannot call yourself an adjective (with an "a" or "an" or "the"). You must find a noun form, and in this case, the noun is Spaniard.

There is a usage of the national denonymic adjectives (that is, words like Spanish, Dutch, Irish) as a noun, that is sometimes found.

However, it's (thankfully) rare, and generally sounds patronising at best, if not down-right rude.

Don't use denonym adjectives as nouns, as you may sound not just grammatically incorrect, but rudely xenophobic.

(Of course, the cases where the adjective and noun match okay, like Australian, German, etc.)

"Spanish" is an adjective.

There are results of "I am a Spanish...", but there must be something following, such as "I am a Spanish teacher/man/etc.."

You can say "I am Spanish", without an article and without something following.

"Spaniard" is a demonym, always with an article. "I am a Spaniard." and nothing following.

When the adjective form and the demonym are in the same form, the article can either appear, in the case of demonym, or be absent, in the case of adjective, and the two options mean the same, such as "I am a Chinese" or "I am Chinese", thought there are subtile differences between them.

Here you have a list over the whole. https://en.wikipedia.org/wiki/List_of_adjectival_and_demonymic_forms_for_countries_and_nations