Example 1 is not correct because of the presence of the past adverbial phrase ten years ago, just like He is in America ten years ago is not correct.
In the sentence, the subject is the clause The first time I went to America; the predicate complement is ten years ago which is also a past adverbial referring back to went, the verb in the subject clause. So, the verb should be was to go with the past adverbial.
The sentenced can be rephrased in the present tense with the use of since.
It has been ten years since I went to America for the first time.
The example 2, I went to America for the first time ten years ago is grammatical.
A similar discussion is here on the ELL stackexchange.