You can buy USB to USB-c cords and adapters, so you do not have to go from usb-c to usb-c only. But USB power standards change over time.
The port you are referring to is the USB-a port, but it is not the “normal” USB port - it is just the first in a standard that has changed and evolved as devices changed and evolved. There are micro and mini USB ports, and usb-c, which is the latest. There are also different iterations of the standard for power. The most recent USB-3.1 standard uses primarily usb-c style ports and allows for up to 100watts of power. This is the type used in the new iPad Pro models.
So while you can buy adapters for different end cables to go from any USB style port to any other, if the port supplying the power cannot supply sufficient for the device, then the device won’t work from that source.
An iphone power brick puts out 5volts at 1amp (confirms to the older USB-1.1 power standard), which is insufficient for the new iPad Pro even though you can get a cable that will let you plug it into the new iPad Pro. The new iPad Pro power brick puts out 18watts of power (within the newest USB-3.1 power standard), over three times that of the iphone’s because the new iPad Pro needs a lot more power to function and charge.
So, yes, you can connect any type of USB port to any other. But the power limitations of some power sources may not be sufficient to charge it, or to charge it while using, or may only allow for very slow charging. It’s not just about getting the two ports connected, it’s about what generation of USB standards is the one supplying the power limited by (or it may even be less than the standard allows, if it simply isn’t capable of generating more, as many laptop ports are limited by the fact they are only connected to a small power source).