I would like to make Diablo 3's gameplay more like Heroes of the Storm's which I am more comfortable with. It looks like rebinding most of the spells to different buttons would accomplish this mostly, but is there a way to change Diablo 3's "move and attack" to HotS's "move and begin auto attacking nearest enemy" system?
No, there is no such system that you are able to take advantage of in Diablo 3. If you click on an enemy that is out of range of your attack, you will walk to them and attack them once.
The closest you are going to get to an "auto attack" is by clicking on an enemy and holding down the button/key, this will make your character attack that enemy until you let go of said button/key or until you run out of energy for the attack.
Alternatively you can hold down the Shift key (by default) to make your character stand in place and then press and hold the button/key of the attack you wish to use.
The answer is no. In Diablo 3, you're in more direct control of your character, while in HoTS and other MOBA's, you control them more like a unit in a strategy game. You can, however, rebind the skill keys to QWER if you prefer, (if so, make sure to rebind the potion key etc. to something else)
However, I have a suggestion for you:
Play the game for a while. Most likely, you'll find that you get used to the control system and why it's well suited to this type of game. Moving from one game to another usually requires a few hours to get used to the control scheme. This is totally normal and something gamers do quite often.