Big Bear Lake is about 100 miles east of Los Angeles — roughly two hours by car on a clear day, though weekend traffic and winter chain requirements can extend this considerably. The main route is the I-10 east to Highway 30, then Highway 18 (the Rim of the World Scenic Byway) up the mountain. Highway 18 through the switchbacks is the most dramatic approach; Highway 38 from Redlands is the gentler alternative and is usually open when 18 is restricted.
There is no public transport to Big Bear from Los Angeles. Mountain Transit runs local services within the Big Bear Valley. Ride-shares are expensive from the valley floor. Most visitors drive.
In winter, always check CalTrans chain control requirements before setting out — chains may be required on Highway 18 and rentals are available in Redlands and San Bernardino if you don't have them.