"The conservatives are getting primaried"
"The conservatives" - this is the subject of the sentence and is a "noun." So you are correct here
"are getting" is a verb phrase. You can combine the verb "be" with a present participle of another verb to form the "present progressive" tense. This refers to an action that started at one moment in the present and is continuing.
"primaried" is a bit strange, but it resembles a verb that is being used as an adjective. If you were to form an infinitive you might say "to primary," but such a verb doesn't usually exist, but it has been created for the purposes of this expression.
So, to summarize:
(noun) + (helping verb + present participle) + (adjective)
This sentence is in the "present progressive tense."
I hope this explanation helps. If you have any other questions please let me know.