There are 79 episodes, but if you'd like a list of ones "worth watching first":
"Balance Of Terror", because you asked about the Romulan War. A follow-up to watch is "The Enterprise Incident".
"Space Seed", because it explains the origin of a main character in one of the movies.
"City On The Edge Of Forever", because it's often rated as The Best ST:TOS Episode, and rates very high on the all-Trek list.
"Journey To Babel", because it introduces Andorians (and Tellarites) to the Trek universe. It also introduced Spock's parents.
"Amok Time", because it gives more insight Vulcan culture.
"The Trouble with Tribbles", because it's just so darned fun.
There's about a dozen more I could recommend, including "Arena", "Mirror, Mirror", "The Devil In The Dark", and "Errand Of Mercy". I'd suggest watching the original pilot "The Cage" before watching the two-part "The Menagerie" because they used a lot of the footage from the former in the latter.
If you search around, or just ask here, you can find many opinions rank-ordering the entire series from best episode to worst. Almost everyone will put "Spock's Brain" in last place, or at least in the bottom five.