"It took 6 hours by car" not "tooks" "I drove the car myself" or "I was the only one driving" or as Maggie says, "I drove myself there" "Now I have just started to examine my patients" is fine. You could use 'begun' and use 'examining' but there is no need to change what you wrote. "Today is going to be a long day!" set expression. You could add 'for me' but it is not necessary. Or you could write something like, "Today it will take a long time to see all my patients' - if that is closer to your meaning. Everything you wrote is understandable... it just needs a little 'tweaking' !

I traveled from a city to another this morning. The drive was 6 hours long. I drove myself there, and that is why I am tired. Now, I have just begun examining my patients. Today is going to be a long day for me. Just a different view on how you can write the paragraph above. Traveling in a car is called a drive Start and begin can be used interchangeably.

I traveled from one city to another this morning. It took 6 hours by car, and I drove by myself, which is why I'm tired. Now, I've just started seeing my patients. Today is going to be a long day for me.