I'm new to Pathfinder and have been familiarizing myself to the rules and mainly spellcasting for different classes. Most emphasize on spells a cleric can and cannot cast seems to be placed mainly on good vs evil alignment but as a Neutral Good cleric can I cast spells labelled as lawful or chaotic or does the same rule apply?
A cleric cannot cast spells opposed to his (or his deity’s) alignment: a good cleric cannot cast evil spells and vice versa, and likewise a lawful cleric cannot cast chaotic spells and vice versa.
There is no alignment opposite neutral, so a cleric neutral with respect to good and evil can cast both good and evil spells, and a cleric neutral with respect to law and chaos can cast both lawful and chaotic spells. A true neutral cleric (of a true neutral deity, if applicable) can cast all of the above.
If this sounds like a big advantage, eh. There are some drawbacks; a neutral cleric cannot take any of the alignment domains (though they generally aren’t very good), and spells like blasphemy exist which target (in this case) non-evil characters, which a neutral character is. So while a true neutral cleric can cast all of blasphemy, dictum, holy word, and word of chaos, he or she is also affected by each of those, while clerics matching the alignment of a given spell are protected from it. And that’s important, because those spells are really, really dangerous, and commonly found on aligned outsiders (e.g. angels or fiends).
The only aspect of the cleric class that really has specifically to do with good and evil, and not law and chaos, is channel energy. As described in that feature, a cleric neutral with respect to those two (regardless of her alignment with respect to law and chaos) chooses which form of energy to channel when he first gains that class feature.