Lake Arrowhead is the mountain day trip I would suggest first from San Bernardino. The distance is short enough that you can leave the valley after breakfast and be among pine trees and mountain scenery surprisingly quickly.
The drive is part of the attraction. You climb sharply from the Inland Empire, gaining elevation and watching the urban valley spread below.
What to do when you arrive
Lake Arrowhead Village is the obvious first stop for shops, food and views toward the lake. Keep expectations realistic: much of the shoreline is private, so visitors cannot simply wander around every part of the water.
I would combine the village with a scenic drive or another accessible mountain activity rather than making shopping the entire day.
Weather changes everything
In summer, elevation can provide relief from valley heat. Winter can bring snow, ice and chain controls. Check current road conditions before setting out. Weekends can be busy, particularly when fresh snow draws Southern Californians uphill.
The dramatic change of environment for relatively little driving is the main reason this trip works so well from San Bernardino.