Riding a road bike for the first time can be an intimidating experience for the reasons your listed. The narrow tires require more balance and lever shfters take time to work through.

The more time you spend on the bike, the more comfortable you will get. I like the idea of taking it to a large parking lot and riding around. Maybe you could put some obstacles out to represent curbs or people and practice avoidance. As for shifting, different people will have different methodology. I base my gear on where I need my cadence and heart rate to be. There are people who go off of power. For a beginner, I would buy a cheap bike computer that can track cadence and find a comfortable zone (75-85 rpm). When you fall below that zone, you know that you need to shift down, making your stroke easier. When you get above it, you need to shift up, making your stoke harder.

Simple guidance. I'm sure others will chime in.

You didn't mention if this is your first bike so I am going to assume you are a complete newb. If you have basic bike knowledge and are just getting into road bikes you can skip the first paragraph.

The first thing you need to do on a new bike is get properly fitted. Most shops will hook your bike up to a trainer and let you peddle for a while and make sure your seat is adjusted properly, bars are at the right angle, etc. Your seat angle and height will have a lot to do with your stability and ability to ride for long durations (both from a comfort and power perspective). If you haven't invest in some clips for your peddles and ride in the trainer with them (you will increase your power by ~30%). Practice unclipping them in the trainer, get used to the feel. I was really skeptical of them myself at first but they are worth the investment and I won't ride without them now. Use your front brake as little as possible or you risk throwing yourself over the handle bars. When you do use your front brakes be very ginger with them. Almost all of your body weight shifts to the front when braking which makes the front brakes a lot more grippy which means it takes a lot less pressure on the front brake lever to get the same stopping power out of the back. I almost never use my front brakes unless I am going down hills.

Regarding the shifting Drudoo's suggestion of getting a computer for your bike is spot on. Its up to you if you want a cheap one of more expensive one. If you are serious about doing triathlons and are able to spend the money it might not be bad to look into purchasing a triathlon watch (DC Rainmaker's web site has great reviews). When I learned my comfort zone in terms of pedal cadence it made the act of shifting so much easier that its now to the point I don't even think about it. Its like when you run you get a certain cadence and its all muscle memory after that.

Pro tip for dealing with stop lights and street crossings. As you approach where you are going to stop, down shift a couple cogs. This make starting off so much easier and greatly reduces your risk of falling over or slipping off the pedal.

Regarding flats, you are going to have them. Its not a matter of IF but rather WHEN. The good news is they aren't dramatic. You will just be pedaling along and realize 'crap, my tire is flat'. You might hear a pop before hand. If you do get a flat most bike shops sell flat kits that come with tools for changing the tire. The CO2 cartridge systems are my favorite as you don't have to worry about carrying a pump with you, but its a matter of preference. Changing your first tire can be a challenge so see if your local bike shop offers training classes or 'lunch and learn' events. If not watch some Youtube videos so you at least know the basics. Its not difficult as long as you know what you are doing. My last flat took maybe ten minutes to fix.

Pro tip for changing flats. Shift your rear derailleur to the smallest cog (highest gear). This makes taking the chain off a lot easier.

Regarding where to keep your hands, keep them where its comfortable for you. I tend to move my hands around a lot to keep from getting sore. I rarely keep my hands in the same position for more than a few miles. The general rule I follow is when going down hill I will get into an aero position but keep two fingers on the brake levers, going up hill I will put my hands either on the flat part of the bar or more usually on top of the shifter hoods so I can down shift if I need to. More specifically, the lower you are in the handle bars the more aerodynamic you are but you produce less power and the higher on the bars you produce more power but increase wind resistance. Also, don't rest your body weight on the bars. Your core should support your upper body, not your hands.

Regarding the idea of riding around in a parking lot I would say it won't hurt but likely won't help unless the parking lot has hills you can practice on. It sounds like most of your worries are with things you have do while riding (hand position, shifting, etc.). Doing circles in a parking lot likely won't expose you to most of this things you express concern with. You would be much better served if you were to get a bike computer so you can watch your cadence and get on a trail with some mild to moderate hills.

Road bikes are a lot different than ''normal' bikes. I have been riding mountain bikes off and on for nearly 20 years. I just switched to road bikes this year. My first few outings were a bit scary but I have gotten used to it and now prefer my road bike over the mountain bike. Regarding the mental aspect of it, it helped me tremendously riding with a group of other riders. They know a lot of the little secrets and tricks (the pro tips above all came from other riders) and most are welcoming of new riders. The rest just comes down to Time In The Seat.

Good luck and welcome to road bikes!

Not much to add to the previous good advice other than ride, ride and ride some more. In no time at all you'll find your comfort level improving.

Don't rush getting clipless pedals, eventually you'll probably want them but for now it's just task loading (pretty much the only time I've fallen was when I stopped and forgot to unclip.....a slow motion fall to the right, fortunately it was only my dignity that suffered)