Ease-up just as you shift and let the chain engage fully before you push hard on the cranks. If you're pushing full power when you shift, you're just asking for the chain plates to pop.
I break rear spokes (usually a one or two a rear), but in my 30 years+ of biking, I've never busted a chain.
When I learned to shift (damn, I'm old), index shifting and contoured ramping was unheard of. It took a deft touch and careful timing to make shifts happen. That's probably what's helped me to not bust chains.
On the mountain bike, I used to go through 3 chains a year, and usually by the 3rd chain I replaced the rear cogset. My middle chainring usually needed replacing every 3 or 4 years. You can tell when the chain begins to skip over the teeth when it's supposedly fully engaged.
Buy chains on sale and learn to replace them yourself.