While I have no experience with crickets, a lot of users say they’re safe as long as they’re kept healthy, and are closely monitored while in the same habitat as the mantis.
Often recommended as a safer substitute would be dubia roaches, which are commonly available at most pet stores, and generally accepted by Mantids as a staple diet.
Aside from fruit flies, winged things are not commonly available other than online. Waxworms, I believe, are sold at PetCo and most pet stores, and they mature into moths. But I’ve heard they aren’t good as a main food source because they’re high in fat. Flies have to be purchased as pupae online and stored in the fridge until you’re ready to give them to your mantis, unless you’re willing to breed them. Although I’ve heard it’s a gross and smelly venture.
Good luck with whatever you choose!