Shifting is a learning process....and I'm still learning
Does your bike have Tiagara or 105?
I have Shimano 105 and have less shifting issues than you describe but I still do have to look down when I do bigger shifts to make sure I'm not stressing my chain/cogs too much :P (i.e. big ring on big ring, etc). When I was starting out on my road bike, I had to look down every time I shifted to verify my shift and what it looked like. Which was kind of dangerous as a beginner because that meant I wasn't always looking where I was going. Then again, I was very cautious about where I rode because I was so unsteady in my shifting. Some people remember which gears they are in but I like to just shift and enjoy the ride :p
Plus I am hyper-vigilant about cars around me--I have to know where they are at all times! So knowing where the cars are takes priority over what gear I'm in! :p
So maybe you'll want to upgrade your front or rear derailleur as you become more proficient at shifting :)
Just curious: did you ride any other bikes with upgraded componentry? Did you notice a difference? :)
One more thing about clipless
Oh! And don't forget that the tension on your clipless pedals is most likely adjustable! That means the amount of force required to clip in or swing your foot out to unclip! If you find the tension that is right for you, you will have less accidents :D:D
I had a more experienced cycler help introduce me to cycling so they loosened the tension for me when I first started out so my feet wouldn't be trapped onto my bike in case I fell because I couldn't detach my pedal :)
But there are all kinds of things I learned as I rode more about my bike :)