Apologetics, rather than being the study of really great ways to apologize, is the rational defense of the Christian faith, and there are five main methods of doing apologetics: Classical, Historical, Presuppositional, Reformed and Cumulative Case. I am just barely becoming familiar with these methods, so this post is subject to updates. By the way, thankyou so much to Eric Chabot for most of the links below.
A review of four of them is here. A book on all five is here.
Classical apologetics starts with natural theology in an attempt to provide good reasons outside the Bible to believe God exists before expecting someone to even consider possible the miraculous bodily resurrection of Jesus (as recorded in the Bible). It is thought that until one believes in God, one has no reason to believe miracles are
possible. So the resurrection (or any other miracle) is not used as evidence of God's existence. Once evidence (from natural theology) of God's existence is believed, good historical reasons to believe the resurrection actually happened will be more readily received. Chart here. A chart of Philosophical Apologetics is here.
Historical Apologetics or Evidentialism:
Whereas classical apologetics is a "two-step" method, historical apologetics cuts to the chase and argues in "one step" from the historical evidence that an act of God occurred in the resurrection. So, instead of first arguing that God exists so that one can then believe the resurrection miracle to be possible, this method is an argument for God's existence. It is limited only to historical evidence for the resurrection. Article here. Another good article here.
A presuppositionalist believes we should not start on common ground with an unbeliever (as in classical apologetics), because such ground is godless. We should only use positive apologetics to show the logical coherence of Christianity, never to build up a case for accepting it, because no one will accept the evidence who does not already accept Christ. Chart here.
Reformed Apologetics (here):
Reformed epistemologists, like Alvin Plantinga, believe that Christian belief is a properly basic belief, like memory beliefs. We don't need to provide evidence to others in order to know our memories actually happened, and likewise do not need to provide evidence to others in order to know the Holy Spirit saved us. This exempts Christianity from being an hypothesis needing to be the best explanation for evidence and arguments, as in our next method.
Cumulative Case (including Constructive) Apologetics:
The Cumulative Case method involves both negative and positive apologetics. Negative apologetics goes on the offense against alternatives to Christianity, as well as going on the defense against objections to Christianity. Positive (or constructive) apologetics, using "inference to the best explanation" or "abduction", (or "hypothesis evaluation and verification" -- Groothuis, Christian Apologetics) takes all types of the evidence into consideration (properly basic, natural theology, historical) and compares competing worldviews to show that Christianity (treated as an hypothesis) best accounts for the evidence: "the existence and nature of the cosmos, the reality of religious experience, the objectivity of morality, and other certain historical facts, such as the resurrection of Jesus" (source). Constructive (or positive) apologetics "builds a case for Christian theism by arguing that Christianity best fits the appropriate criteria for worldview assessment," (Groothuis, Christian Apologetics). Article here.
I am most comfortable with this last method, because I can see how there is historical evidence supporting the resurrection and therefore God's existence, but I also think that our hunger for true meaning (a clue to the existence of objective morality) should be the presupposed evidence we start with, for it is that hunger that Jesus satisfies when he demonstrates God's contra-conditional love in switching perspectives with us on the cross. Also, I think the historical method uses "inference to the best explanation" when it rules out competing naturalistic theories in favor of the resurrection being the best explanation of scholar-accepted historical facts. Finally, I think if you retain the arguments from classical apologetics along w/ what I've just covered, it makes for a good avalanche. :)