What did the Judaizers believe?

There are quite a number of mentions of their ideas as they are effectively refuted or warned against particularly in Paul's letters. Acts 15, where the whole question of legalism was discussed, especially in relation to the Gentile believers goes into some detail on the matter.

1And certain men which came down from Judaea taught the brethren, and said, Except ye be circumcised after the manner of Moses, ye cannot be saved.

From Paul's letters it is obvious that just circumcision alone was not the only issue in other places, but it certainly symbolized what they stood for in terms of performance of works to earn salvation rather than salvation by grace through the finished work of Christ.

They thus sought to return (if in fact they had ever left it) or add works to faith in order to please God and earn salvation. In this sense they were returning back again to Judaism, rather than moving on to the full personal trust in the finished work of the savior Jesus Christ.

Galatians also deals at length with this whole issue as does Romans. Paul evidently understood, being a former Pharisee, how the Judaizers considered the keeping of the Law of Moses, as symbolized by circumcision, as being an additional and essential part of faith.