This Chateau housed many royals of Navarre, Béarn, Gascony, etc. with the most famous being the…read morepopular King Henry (Henri) IV who was born there in the 1500s. He was baptized Catholic, grew up Protestant (Huguenot) because of his mother's influence, but later converted to Catholicism as King of France. He is popular for bringing peace in light of the Wars of Religion, and he survived several assassination attempts until one got him in Paris in the early 1600s.
The exterior of the Chateau is easily accessible from downtown Pau as it's at the end of the Boulevard de Pyrenees. A tourist train named after Henry IV stops at the Chateau and other locations in Pau (like a hop on hop off bus). The exterior, including the Jardin, is totally free.
If you're not proficient in French, I personally would forgo the inside tour, unless you're very passionate about tapestries, the history of Henry IV, the history of Navarre, art and furniture from the 1800s, etc. Although the Chateau is much older, mostly the current interior decor has been in place since the mid-1800s. The 1-hour tour with a guide accommodates about 30 visitors and starts about every 30 minutes. The tour is given only in French although some signage is available in each room in Spanish and English. You cannot enter the chateau on your own as you must have your tour guide usher you room-to-room. About a dozen rooms are shown, including dining rooms, bedrooms, staircases, current art installations, and more.
You may be able to enter the gift shop without doing the full tour.