Anthony - congrats on your new trainer! I too initially used the Kurt Road Machine which is a classic trainer, meaning that the zwift program does not control the resistance on your trainer. What follows is based on that assumption, so take that into consideration.
For classic trainers, Zwift has what is called ZPower, which uses a unique algorithm to calculate your estimated power output based on your weight, speed, etc. There’s no need to calibrate anything. When the gradient increases, you won’t feel any difference in resistance, although your speed will decrease if you don’t increase your cadence/effort. In other words, it won’t get any harder unless you go faster. That’s the big difference between classic and smart trainers, where the resistance is much more realistic.
You should be shifting during regular rides, just like you would if you were riding outside, unless you’re content to keep the effort consistent throughout your ride.
Zwift is designed to be used with PCs, laptops, Macs, and iOS systems. As far as I know, Zwift for Android may not yet be available. I could be mistaken, however. Recommend you check the “Getting Started” link on zwift’s homepage for further guidance on system requirements. The Knowledge Base is also a handy reference.