Dish Network Corp. unveiled a new $20-a-month version of its Sling TV streaming service that will include channels from 21st Century Fox for the first time, setting up a more robust competitor to traditional pay TV.

The new package, which launched Wednesday in beta, will allow for three simultaneous streams on different devices in hopes of better catering to families with multiple viewers. It will be anchored by Fox channels like FX, Fox’s regional sports networks and the Fox broadcast network in 17 markets initially, with...