On first glance, it may appear like there's not much to see at Jongmyo Shrine. Its main raison d'être is to hold the spirit tablets of the Joseon kings and queens - but these are all hidden away in spirit chambers in the main buildings where no tourists will ever see them. On top of that, the architecture is much more muted than the nearby royal palaces, so it's less of a spectacle. And, at the time of my visit, the main hall (Jeongjeon) was covered by hoardings while it undergoes a five-year renovation (due to finish in 2025).
However, as I learned more about the story of the shrine, I gained a new appreciation for its heritage and how it plays a part in Korean culture. As the guide described it, Jongmyo Shrine is "a symbol of the Korean people’s thought". The Confucian principles at the shrine's core are quite interesting and seeing a place where they are put into action is part of exploring the country's guiding philosophy.
There's also something quite beautiful about the architecture once you accept that is is deliberately less flashy, presenting a more respectful design for a place that honours the dead. The landscaping of the parklands in the shrine creates a tranquil setting away from the bustle of the city.
If you visit on a weekend, you can wander around at your own pace. On a weekday, you need to take a tour, which lasts for about an hour (the shrine is closed Tuesday). I really enjoyed the tour and learned a lot, with the guide giving a good overview plus interesting details. There were only a dozen people in the group, so there was ample opportunity to ask questions and chat as we walked between the different locations.
It makes sense to combine Jongmyo Shrine with a visit to the other nearby WHS of Changdeokgung Palace. I would also recommend buying the Royal Palace Pass (for the crazily cheap price of 10,000 won) which gives you entrance to these two World Heritage Sites, plus three other palaces in Seoul.