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.