Nausicaa of the Valley of the Wind and Castle in the Sky are the most similar in feeling to Princess Mononoke. High fantasy, adventure, against all odds. (This is my favorite batch of Ghibli / Miyazaki / Takahata films.)
Spirited Away is an otherworldly coming of age treasure that cannot be missed. Totoro and Kiki's Delivery Service feel similar. Whisper of the Heart is this too, just aimed a little older. (And if you like that, check out Your Name by Shinkai or Wolf Children by Hosoda - not Ghibli films, but very deserving of a watch if you like this sort of film.)
Grave of the Fireflies is a very poignant anti-war film.
FWIW, I bought the old Miramax domain name for Princess Mononoke and kept the website online [1]. It's one of my favorite films. I wish Miyazaki made more films like Mononoke and Nausicaa - they're sublime.
On a more serious note, the book is quite good too. The book doesn't describe the castle as a giant steampunk thing walking around in the hills. It's actually just a plain 4 sided stone building that hovers around a little over the ground, no legs. So in this regard the movie is better IMHO. But in the book there is another location the door opens to and the stuff we see in there changes the whole story. In the movie we spend some time trying to figure out what version of the characters, Howl especially, are the real version. The book turns this up further with the additional location.
At the end the take away is quite different between book and movie. Both good in their own ways.
I love the ending
As far as I am aware, most anime nerds say that Grave of the Fireflies is the best movie from Studio Ghibli (for this movie, Hayao Miyazaki was only producer and not director). But be aware that Grave of the Fireflies is a rather sad movie.
It was made together with My Neighbour Totoro as a double feature. Only Totoro could rescue me!
Actually that's a good way to think of it, compress SL into a anime length movie and you have the emotional hit of Grave.
But not for a second regrettable.
As sibling comment notes, have My Neighbor Totoro ready to go afterwards to put you back together.
It's Schindler's List in a few important ways. 1) it's about WWII, 2) It's a gut punch, and 3) It's one of the greatest movies you only need to see once, and likely will only want to watch once.
I've told the kids they can only watch that movie once they're old enough to drink.
If there was, wouldn't you want to know?
So, 6 months old? (When I grew up, whiskey was a common cure for toothache, gripe, or any other bother infants had...)
Spirited Away, Ponyo, Howl’s Moving Castle, and Nausicaä of the Valley of the Wind.
I don’t know that I view movies as “best”, since those are all quite different stories.
That said, I've liked pretty much every film they've made except Kiki's Delivery Service (I found it a bit dull) and Howl's Moving Castle (it just didn't work for me). They're pretty much all worth watching at least once.
Porco Rosso seems to get ignored frequently. It’s fantastic as well.
Edit: Also, Jiji (Kiki's cat) played by Phil Hartman has a completely different character from Jiji in the Japanese version.
For a pre-teen audience, Nausicaä of the valley of the wind, Castle in the Sky, Totoro, basically all the stuff before Princess Mononoke are great films.
I see Princess Mononoke to be a film for people coming of age, where it's time to realize the binary view of the world is overly simplistic.
Just don't bother with the dubs. Hollywood just couldn't get the right voice actors to act with the right voice to fit the setting and the theme. For example, the main characters dub for Castle in the Sky felt older than the original (or Cantonese dub, which was my first introduction). Pazu was supposed to be an orphan, a child worker in a frigging mine. While the film doesn't directly call attention to that much, it hints at it by having the original voice to be too young to even crack yet.