The three degrees of glory are the telestial, terrestrial, and celestial. Each kingdom is on the order of the law it represents: Telestial law is the law of the world, effectively lawlessness (in the scriptures, this is compared to the glory of the stars), Terrestrial law is a "middle" law (compared to the glory of the moon), and the Celestial law is God's law (compared to the glory of the sun).
There's not exactly a checklist of requirements to pass off before inheriting a higher kingdom.
In the Book of Mormon, Alma teaches that individuals are resurrected with (he uses the term "restored to") a physical body comparable to the law they lived. In other words, individuals who live a telestial law will be raised to a telestial body and thus will inherit that kingdom.
A revelation to Joseph Smith and Sidney Rigdon opened up the windows of heaven to them, which is detailed in Doctrine and Covenants 76, and it explains the status of those in each kingdom.
Evidently, those in the telestial kingdom will have not chosen to receive a testimony of Jesus, they rejected the gospel, and are "thrust down to hell" (see this question to learn more how this phrase is interpreted), yet they did not deny the Holy Ghost (in other words, they did not come out in open rebellion against God). They will be resurrected last.
Those in the terrestrial kingdom will have received the Spirit and gospel (since LDS theology teaches that everyone will have the opportunity to receive it at some point, either in this life or the next), but did not receive its fulness, by their choice.
Those who inherit the celestial kingdom have faithfully accepted the fulness of the gospel of Jesus Christ, having: entered into baptism by immersion for the remission of sins and received the Holy Ghost by the laying on of hands by those in authority. Within the celestial kingdom are all those who are pure and holy, and even those who have achieved exaltation, God's greatest and eternal blessing to His children, that they become like Him. These souls will be resurrected first.
Now, your question about "hell," being the temporary spirit prison or the eternal darkness. In LDS theology, after death, there is a "spirit paradise" and a "spirit prison." Often it's taught in Sunday schools that they are separate places, like two buildings across the street, but the more I study it, the more I'm realizing that it's two states of being.
Spirit prison is a state for those who already chose not to accept the gospel or would reject it when it is preached to them. Spirit paradise is a state inherited by they who lived the gospel, or at least its teachings, whether or not they received it. Those who did receive it, though, preach to those who did not. Peter mentions this work in 1 Peter 3:18-20. D&C 138 also talks about this work in detail. These spiritual states are only temporary, while the resurrection is awaited. To "escape" that hell, I suppose one would have to accept the gospel, implying repentance and some payment for sins...
Finally, the eternal sense of the word hell is reserved for those who reject God, refuse the gospel, and deny the Holy Ghost. It's understood that these will be relatively few in number, but the sons of perdition, having departed in a sense the family of God, will not receive any glory.
Some people use terms I've used in my answer a little differently in some contexts, but the differences are minute.