I've just transferred to another elementary school after my third job as a teacher. (We would say job instead of career. A career is really what you do overall. So your career is that of a teacher. You have had three different jobs teaching though, within your career. Hope that makes sense.)

Just two days have passed since my new life started,

Now I am very exhausted! (Just a note that in English we don't start sentences with But or And)

For sure, I'm still not accustomed to be in a new environment-relationship among coworkers, allocation of goods, the whole system. (While it works, "For sure" is not really a good way to start a sentence. It would be better to say something like, "I'm definitely still not accustomed")

I am in charge of jobs that are much tougher than before.

I need to do my best in my work, but at the same time, I would like to do my best in utilizing my private time freely, very well! (very well at the end is redundant ant not necessary. This sentence works but doesn't flow quite like regular English. Perhaps another way to say it would be, "I want to do my best at my job, but at the same time, I would like to enjoy my private time as much as possible")