As others have said .... well-adjusted bike shift just fine under load, particularly in the back. I do recommend easing up for a fraction of a pedal stroke just as the chain is floating between cogs, but that just makes it easier ... with modern derailleurs you can shift while pedaling full gas.

Plenty of good climbing technique described here, but all of it will work better when the bike works. I assume your cables have stretched. Take it back to the shop for a tune-up. have them check the brakes and the spoke tension while you are there.

Great.