A priest is what in most translations is called an elder. The word priest in English comes from the Greek word for elder, πρεσβύτερος, or presbyteros.
The New Testament "instruct[s] us to have priests" in Titus 1:5, appoint elders in every town as I directed you. There are also hints of it elsewhere; priesthood is conferred by ordination or laying on of hands, as referred to by Paul in 1 Timothy 5:22 and 2 Timothy 1:6.
It's in Titus, right after the above verse, that certain qualifications are stipulated for priests:
if any man is blameless, the husband of one wife, and his children are believers and not open to the charge of being profligate or insubordinate. (Titus 1:6) Husband of one wife, i.e., married no more than once. This doesn't disqualify unmarried men because Paul was unmarried.
Overseers is a somewhat literalistic translation of ἐπίσκοπος or episkopos which comes from the prefix epi, over, and skopos, one who sees. We would call such a person a bishop in English (in Latin the word is episcopus, much closer to the Greek.) In parts of the New Testament, the terms presbyteros and episkopos seem to be somewhat interchangeable, but in other areas (such as in Titus) there is a clearer distinction. Bishops assumed, over time, a higher position of authority, overseeing the priests entrusted to their care (and of course their flock).
It sounds like you have a Protestant friend who is trying to nail you on priests and what Scripture has to say about them. The root question is whether it is necessary for everything to be proven by the Scriptures in order to be valid.
While the written Word of God is "given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work" (2 Timothy 3:16-17), there is nothing in Scripture (or Tradition) which demands that everything be explicitly provable by the Scriptures to be accepted. Saying you can reprove or correct someone who is in error using the Scriptures is very different from saying that everything you do or believe has to be provable by Scripture or it's invalid. It is the Church, not the Bible, that the is the pillar and foundation of the truth (1 Timothy 3:15) and we must look both to Scripture and Tradition (2 Thessalonians 2:15) for our beliefs.
The Church developed over time according to its need; what served a small underground church for a few decades doesn't necessarily serve a 1 billion person behemoth 1,900 years later. Think of the difference between the way a ten-person startup is run and the way a 100,000-person multinational corporation is run. There are going to be different policies, organizational structures, hierarchies, customs, and so forth, and that's OK.
I hope this helps.