Priesthood of All Believers, also Women.

 In the Old Testament, priests were mediators between the Hebrew believers and God. This means they were intercessors. Jews also brought sacrificial animals to the priests, who then offered the animals to God on behalf of the people.

As Christians, we are all called, men and women, to pray for each other. God hears our prayers because of the blood sacrifice of His Son Jesus on the cross.

Priests were also spiritual teachers. They studied the scriptures and taught the people how to please God. Since Christian men and also women are appointed as priests in the New Testament, Christian men and women are appointed to study and teach the Bible.

However, some will object that Paul instructed women to learn quietly and submissively. 

This does not mean that women cannot be teachers. It means that women must learn before they are qualified to teach. Of course, men must also not start teaching before they have been trained as disciples.

A woman should learn in quietness and full submission.
I do not permit a woman to teach or to assume authority over a man; she must be quiet. 1 Ti 2:11–12 NIV

Here Paul was addressing a situation in Ephesus, where people, including certain women, were teaching false doctrine. They were new converts who had previously worshipped the pagan Goddess Artemis, a virgin Goddess who was supposedly a source of wisdom and magic power.

There were many unmarried virgin women who served in the temple of Artemis and were respected as spiritual teachers. When they became Christians, apparently some of them continued to dispense spiritual teaching, but their doctrine was mixed with the pagan ideas they had believed before their conversion. Paul wrote to Timothy, telling him to correct this situation.

As I urged you when I went into Macedonia, stay there in Ephesus so that you may command certain people not to teach false doctrines any longer or to devote themselves to myths and endless genealogies. Such things promote controversial speculations rather than advancing God’s work—which is by faith.

The goal of this command is love, which comes from a pure heart and a good conscience and a sincere faith.

Some have departed from these and have turned to meaningless talk.
They want to be teachers of the law, but they do not know what they are talking about or what they so confidently affirm. 1 Ti 1:3–7 NIV

What were some of the false teachings that Paul wanted to correct?

Pagans in Ephesus believed that Artemis came down from heaven, and that women were created first. So Paul summarises the Genesis creation story, in which Adam was created before Eve.

For Adam was formed first, then Eve.
And Adam was not the one deceived; it was the woman who was deceived and became a sinner. 1 Ti 2:13–14 NIV

So does this mean that all women are more easily deceived than men? That is reading a lot into a few words, and the bible repeatedly tells us about both men and women who are deceived.

But didn’t Paul also specifically say that mature women should teach younger women?

Likewise, teach the older women to be reverent in the way they live, not to be slanderers or addicted to much wine, but to teach what is good.
Then they can urge the younger women to love their husbands and children, Tt 2:3–4 NIV

If women should teach women, this does not mean that they must only teach women.

Since women, as well as men are called to be priests, then women must also study to equip themselves to teach the Bible. 

Paul sent Phoebe to deliver his famous letter to the church in Rome. Part of this assignment was to read the letter to the church and answer questions. This was a priestly teaching ministry.

Paul tells the Roman church that Phoebe is a deacon, not a deaconess. This indicates that Phoebe was a minister of the Gospel. The Greek word DIAKONOS is masculine. In the Gospels, it can refer to household servants, but in Paul’s writings it always refers to a minister of the Gospel.