What do the different Republican factions believe in?
When discussing US politics to friends overseas the one issue which I always requires explanation is that America is a two party system. Any third party which fractures a larger party nearly inevitably leads to massive power losses for that party and the opposition swings into power. Two past occurrences of this were the 1860 election, won by the upstart GOP (won with 39% of the vote) and 1992 won by Bill Clinton with 43% of the vote. Most other nations, with parliamentary systems have multiple parties which allow a closer association with the individual people. For example, Japan has 4 or 5 major parties, Australia has four, New Zealand has four. The US political system forces all who wish to be represented into two big tents. For example, libertarians have little common ground with modern Democrats, but are often at odds with parts of the GOP as well. Most whom I know, are at least happy to have Rand Paul as a Senator, who can speak for their aims. This was not always the case.
The five major wings of the party can each be explained by which of five questions they consider the answer to be paramount. Libertarians are the easiest to identify because they believe in maximum liberty, hence the name. Government is an evil, a necessary evil, but evil. While libertarians aren't anarchists they think the government mandate from the masses should restrict government to certain specific rights and roles. Immediate national defense, regulation of the quality of products, interstate commerce, etc.
Christian conservatives are a mirror image of the Democratic secularist wing. Religious freedom does not end at the walls of the church, but extends throughout their life. Living one's life by the rules of his/her own choice of church is, to CCs, guaranteed in the Constitution.
The establishment, moderate and Tea Party wings are all described on one their key goal. The establishment wants to retain power (for themselves) and will abandon anything for that one goal. The moderates seek to govern and to attempt to move towards common GOP principles on social, economic and regulatory issues, but not at the cost of governing. The Tea Party does not make that compromise. They would rather move, even if symbolically, towards GOP goals than cooperate and govern.
All but a few members of the GOP believe in the following common goals. 1) Individuals should take first, primary responsibility for their own health, prosperity and freedom. 2) An individual should be able to rise, or fall, to the level of their work output, their intelligence and their commitment, 3) a free market is the most efficient system for distributing almost all goods. 4) government regulation should be limited to quality and safety. 5) freedom of religion is a founding principle of the USA and means more than just being able to build a church or worship there. 6) government is a necessary burden upon society, to be paid for by all in proportion to their ability to pay. 7) Veterans and current members of the military are a protected class, often, the most protected class, which because they choose to serve for the protection of all, deserve greater consideration than any other class of people. 8) Healthcare of good quality, should be accessible to all, but those whom society values higher, should have access to a higher quality of care. 9) Education follows healthcare.