If you're new to Ghibli, my recommendations are as follows:
1. Princess Mononoke
A fantastic piece of eco-criticism with unusually complex and morally ambiguous characters set during a fictional iron-age. IMO the best animated film ever.
2. My Neighbor Totoro
This one is really a classic. Totoro tells the story of two young girls move to the countryside with their father after their mother falls ill, only to discover that there is a wondrous world hidden there. Story isn't really the right word here. This film is more about experiencing a place and a mindset than it is plot-driven. Truly magical.
3. Grave of the Fireflies
Directed by Isao Takahata, Grave takes a more serious tone than most of Miyazaki's films. Set in WWII-era Japan, this tearjerking, heartstring-tugging film follows a boy and his young sister during and after an allied bombing campaign.
4. Porco Rosso
This one doesn't get as much attention, but it's one of my favorites. It's Miyazaki doing his best take on Hemingway, with some magical realism and fighter planes thrown in for good measure.
5. Spirited Away
By now, every one and their mother has seen or heard about this visual stunner. Still the only foreign film to win the Oscar for Best Animated Film. If you haven't seen it, go do it.