I see many Scriptures that talk about this "age" and the "age to come." What are ages and how many are there?
Regarding the ages mentioned in Scripture, there are at least three and possibly four ages referenced in Scripture.
First, there is the age prior to Christ's appearing. Peter describes this age of waiting for Christ in the introduction to his first letter (1Peter 3-12):
In verse 5, Peter refers to our salvation in Christ has been revealed in the "last time." In the Greek, the words are "eschatos kairos," which translated literally means "the end of the age." The salvation we have now in Christ revealed is a salvation revealed in the last age. So, ages are measured, according to Scripture, with respect to God's plan of redemption.
The age before Christ's revealing is the first age, which then gave way to our current age called "the last times" or the Church age. Notice that Peter says in verses 10-12 that the prophets were serving us in that earlier age by revealing the truth concerning Christ and the age to come, things that angels long to look upon. Paul echoes this in his letter to the Colossians (Col 1:25-27):
Other verses in the New Testament (including the ones you quoted) speak of yet another age to follow this one. Again, ages are defined according to events in God's plan of redemption, so this future age will complete God's plan of redemption. According to Scripture, we know this future age is the age of Christ's kingdom on Earth, the Millennial reign of Christ described in Revelation 20. This age will, in turn, give way to a fourth age called the Eternal order. Paul describes this age in 1Corinthians 15:25-28:
Paul gives us glimpse of what follows the 1,000-year reign of Christ, when he describes the age to follow as a time when God becomes "all in all." John also gives a tantilizing description of this future age in Revelation 21 and 22.