Which sentence is punctuated correctly?
a: my father loved talking about history, particularly: the civil war, the cold war, and the revolutionary war.
b. there were several scientific processes i wanted to learn about such as, photosynthesis, diffusion, and osmosis.
c, the only states that marina had yet to visit were: colorado, wyoming, north dakota, and rhode island.
d. when studying mythology, i learned about several gods and goddesses: zeus, apollo, aphrodite, hermes, and hercules.

sentence which is punctuated correctly is,

When studying mythology, I learned about several gods and goddesses: Zeus, Apollo, Aphrodite, Hermes, and Hercules.

Punctuation are used to organize a sentence additionally to create more clear sense and stress in a sentence. In the above sentence, The punctuation Colon (:) organizes the sentence properly when placed rightly. Colon is used preceding any list, explanation, description or definition.  

Further,

Colon is one of the punctuation which is used to:

1. Lay emphasis

Example: He loved just one: Bruno, his dog

2. To present dialogue ( we write the name of the speaker and then insert colon followed by his/her dialogue.)

Example: Rita: How are you?

Mary: Fine, and you?

3. Clarify composition titles ( to explain parallel ideas colon is used in compositions like lectures or book titles), and

Example: Rousseau’s Dog: Two Great Thinkers at War in the Age of Enlightenment

4. To introduce text or lists.

Example: Three types of Subordinate or Dependent Clauses are: Adjective clause, Adverb clause and Noun clause

the anecdote makes readers feel sorry for the man’s lack of heritage, then reminds readers that the man has an equally valuable heritage.