You need to put both on. Primer then under coat. De - nibbing between coats. 2 - 3 coats of undercoat.. You can get a all in one primer undercoat. Dulux trade do a good one in my opinion. 2-3 coats... Dulux trade gloss is also good.
One thing I woud have done before assembly is to have given the parts that you will not be able to get at now a coat of primer to aid in protecting... Also you could use a preservative for extra fence life but you need to find a clear one that will take a painted finish.
The Dulux paint system (weathershield ?) is very good and comprises a preservative followed by primer undercoat and finished with gloss.
Expected to last in excess of ten years before requiring maintenance.
All the other major manufacturers do their own version