Barrie described the difference rather well. As for your question on tenses, the expression get to know can indeed be used in a variety of tenses:
I just got a new job. I expect that I will get to know my new boss much better over the next few weeks.
I'm liking my new job. I'm starting to fit in as my coworkers are getting to know me better.
Now that I'm retiring, I'll miss this place. I got to know a lot of interesting people over the years.
I was a little sad when I put in my retirement papers; I had gotten to know a lot of interesting people at that job. I still miss them sometimes.
Note how, in the first example, get to know is different from meet. The new employee may have already met her new boss, and could perhaps recognize the boss in a crowded store. But knowing a person implies more than that. When you know someone, you understand their personality and quirks: you understand their sense of humor, you know what irritates them, you can tell when they're feeling sad. You know what sports teams they like, you know what they like to do on the weekends, you know their favorite movies or books or television programs. Getting to know someone is the process of acquiring that knowledge; it usually takes months or years.
You can also know someone and be getting to know someone at the same time – if you've learned a lot about that individual, but are also still learning new things about that person on a regular basis.