Riding a road bike is not that different, except you will be going faster, a bit. I assume when you say you have a mountain bike, you mean you are riding it on the road, and not on the trails. That is how I started, very similar to you. I was fit and semi athletic from gym stuff and my DH and son were into riding. My first ride, 6 miles, on a mountain bike with slick tires, on the road, just about killed me. My DH had to put his hand on my back and push to get me up a short hill! I rode this bike for a year and a half and then I got an entry level road bike. While the shifters are in a different place, the actions are pretty much the same. Getting used to the skinnier tires did not take me that long, either. And, I have to add, that I am not super coordinated and my spatial skills kind of suck. So, if I were you, I'd get used to riding, i.e., shifting, cornering, climbing and descending on your mountain bike and then, when you are comfortable, you can transfer those skills to a road bike. I lead rides with a lot of beginners and I'd say that I am always surprised at how many people have no intuitive sense of how to use their gearing and think that it means they are "tough" if they don't shift down to easier gears while climbing.
As far as HRM, I stopped using one years ago. It never worked, always got interference from other people's monitors, and it only told me what I already know, that my HR runs higher than it should. But, it's always consistent, and my RHR is low.