What are you doing tomorrow?

Question 1

We also use am/is/are + -ing (present continuous)

Answer 1

for the future (tomorrow / next week etc.)

I am doing something tomorrow = I have arranged to do it, I have a plan to do it

Sophie is going to the dentist on Friday.(= she has an appointment with the dentist

…Andrew is playing tennis tomorrow.
…I’m not working next week

Question 2

We use the present continuous (I’m staying /are you coming etc.)

Answer 2

to say what somebody has arranged to do

…I’m staying at home this evening. (not I stay)
…Are you going out tonight? (not Do you go)

But we use the present simple (start, arrives etc.) for timetables, programmes, trains, buses etc. :

…The train arrives at 7.30.
…What time does the film finish

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present continuous (usually for people)
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present simple (for timetables, programmes etc.)