I usually go to Starbucks with my tutor to have my class while drinking coffee, and chat with my tutor in Chinese to practice my speaking.

I think it's more natural to say "have class" since it's implied that it's your class. Also, when you have a compound verb (2 or more verbs with one subject), you don't need a comma separating the clauses with the different verbs (this is a very minor correction!)

But our last lesson was different from what it was usually is.

I'd use "is" instead of "was" because it sounds like you will continue to have your normal lessons, so this unusual lesson is more of an interruption of those lessons than the end of the lessons (which means that you shouldn't refer to a usual lesson in the past tense)

(She needs to go to many places to teach) mMy tutor said, because there are already confirmed cases in Beijing, weit would be better to have our lesson on WeChat this time.

You can also say "we would be better off having our lesson on WeChat this time."

Although I don’t like online lessons (I don’t think it has a better's more effective than in -person lessons), I think it would be better to listen to my tutor because she is different from me because she knows the specifics of the COVID-19 situations better than me.

I'm assuming that your tutor is different because she knows more about the COVID situation, so it's not necessary to say that she's different than you. If you say she's different because of her profession, I would say that more directly.

Besides, she refuses to go to public places during the epidemic, which makes sense.

Adding "which" indicates that you're talking about your tutor not going to public places

The responsibility she shows to Beijing people is worth me to admiradmirable to me.

You can also say "I should admire the responsibility she shows to Beijing people" or "It's worth admiring the responsibility she shows to Beijing people" if you want to emphasize the worthiness.

After the online lesson, I originally planned to go to Dongzhimen to watch a movie, but I was a little lazy that day, and I just stayed home watching a movie, browsing the network.

When you combine two independent clauses with "and," you need a comma after the first clause