Navarra is quite big. Only 600,000 people live there, and around half of them live in or around the capital, so that's a lot of open space. I haven't visited even half of the natural sites in the province, but here are some of the places I've been to over the last year.
The Baztan Valley
From Pamplona: 1 hr
To see: Natural park, witches caves
Famous for: Cheese, murder mysteries, Camino de Santiago.
With its rolling hills dotted with sheep and its traditional white stone cottages, the Baztan valley is a pretty place. These days it receives a lot of visitors due to the increased traffic on the Camino de Santiago, and because of a popular series books set in the area. The first instalment of the the Baztan Trilogy (a murder mystery written by Dolores Redondo was recently released as a film.
During my visit, I bought cheese from a vending machine, and walked a long way (to nowhere in particular).
The Irati Forrest
Location: North East Navarra, bordering France.
Towns: Orbaizeta, Fabrica de Orbaizeta, Ochagavia
To see: Trees, lakes, rivers
Famous for: Autumn leaves, chestnuts, mushrooms
According to Navarra's tourist information, it is Europe's second largest and best preserved beech and fir forest.
It's probably best known, and most visited in Autumn, when all the leaves change and people forage for forest delicacies.
La Sierra Urbasa-Andía
Location: West Navarra
From Pamplona: 50 mins
Towns: Alsasua, Estella
To see: Natural spring, mountain plateau
Famous for: Beriain mountain, birds of prey, Mirador Balcon de pilatos, Nacedero del Urederra
The source of the Urederra river is another tourist hotspot to in the south of the park.
From Pamplona: n/a
Towns: Lekunberri, Andoain
To see: Tunnels, forests, towns
Famous for: Hiking, cycling
They work well as small towns connect up the line along the way (good for rest stops), and it is often possible to arrive via public transport.
Las Bardenas Reales
From Pamplona: 1 hr
To see: Desert, Stone pillar formations, banded colours
Famous for: Military zone, unique flora, rock formations
Overall, it is a worthwhile (if sweaty) day out.
I'm sure I've missed out some of the province's best loved sights, but hopefully, this gave you a little idea of what's of offer in Navarra, apart from watching people run away from bulls.