Evangelicals have a commonly held belief that the priesthood of all believers somehow replaced ministerial priesthoods. However both ministers and believers existed prior to the Mosaic law as shown in previous answers above. This was also the case in the church under Moses. No scripture in the New Testament states that ministerial priesthoods ended at the cross. In the following passage, it merely says that there was a "change," not an end:
For the priesthood being changed, there is made of necessity a change also of the law.
To any serious student of the Bible, there is no doubt that ceremonial practices such as animal sacrifices ended at the cross. Feasts and holidays that governed the nation of Israel are not officially observed by Christians in this age. The New Testament clearly did not perpetuate these aspects of the Mosaic law.
However at the same time, Christians continue to believe in the Great Commandment. They also affirm the principles of the Ten Commandments. For many years, they used to observe one day a week as a Sabbath day of rest. Each of these practices has its roots in the law of Moses.
When the Hebrew epistle says there was a change in the priesthood, it is entirely appropriate to allow for the possibility that ministerial functions continued as well. There continued to be ministers in Christ's church, but the specific functions that they performed changed when Christ offered the last, great sacrifice upon the cross. There was definitely a change in both priesthood and law after Christ's crucifixion.