There are lots of places to hike in L.A. County. First and foremost is the San Gabriel mountains. There are dozens of places to hike there, offering varying levels of challenge. Heninger Flats, Chantry Flats, and Mt. Lowe are all near Pasadena, and offer relatively easy hiking, while also providing access to more difficult trails, as well. Higher up, you can hike up to Mount Wilson, where there are several trails. Here is a link to some of the trails in that area:
Hikes in the SGM, Mt. Wilson Region
Note: the website above is old and shows a closure of the Chantry Flats area due to a fire in 1999! I would certainly hope it isn't still closed from that event. My favorite hike is number 47 from Chantry Flats up the West Fork of the San Gabriel River, and back around to Altadena through Arroyo Seco.
A little higher in the mountains is the Chilao area:
Hikes in the SGM, Chilao Region
Here are some in the Tujunga area:
Hikes in the SGM, Tujunga Region
Here are some in the northwest corner of the county:
Hikes in the SGM, Northwest Region
These trails are in the Islip Saddle and San Gabriel River areas, toward the eastern side of the county, and possibly even into San Bernardino County:
Hikes in the SGM, Mt. Islip Region
Hikes in the SGM, San Gabriel River Region
These links are all from this site, which covers the San Gabriel Mountains:
Select a Hike By Location Using Clickable Map
Other places to hike in the L.A. county area are the Verdugo Hills above Burbank and Glendale, as well as Griffith Park the the Santa Monica Mountains.
This site lists most of the trails in the Verdugo Hills:
LA TRAILS - Verdugo Mountains
Here's one for Griffith Park:
LA Trails - Griffith Park
That should be enough to get you started, but if you still want more, try looking up "hiking in the Santa Monica Mountains" on Google (check for areas around Mulholland, Topanga Canyon, or Malibu Canyon). That will cover most of the mountainous terrain around Los Angeles County, but there are still a few areas I didn't mention, mostly because I'm not familiar with them.
I am certain that you will have plenty of opportunity to find great places to hike in and around L.A. County.