I noticed that some spells with the combo of Divine or Summoning, creates the ability to cast new spells. Question is, how do you get divine or summoning, are they certain nodes?

Certain units start with them(divine is mostly priests and stuff) summoning is summoners conjurers and ofc a ton of special units can start with any combination of these + sorcery(the last magic branch).

Basically, "Divine," "Summoning," and "Sorcery" are tags that identify a unit as being capable of casting Divine, Summoning, and Sorcery spells, respectively. For instance, the "Sun II" promotion grants "Whitefire" (Divine), "Summon Sand Lion" (Summoning), and "Blinding Light" (Sorcery). That means a divine unit with Sun II (such as a Vicar) casts "Whitefire," a Conjurer with Sun II casts "Summon Sand Lion," and a Mage with Sun II casts "Blinding Light."

Take a look at this spell chart, to get an overview which kind of common units (Heroes aren't included) is capable of casting which spells.

Spell Chart (http://phrasr.com/ffh/pedia.php?v=spell_chart_graphical)