I am completely new to assembler and I am starting to write in assembler 8086-8088 with tasm.

I have the beginning correctly. I need only the coding for reading two digit numbers. So far I know that you have to multiply the first by 10 and add the second.

Please help me get started.

Please explain again because I don't understand you?

I have to write an assembler program with 8086/8088 that reads in two 2-digit integers, one keystroke at a time and then does Euclid algorithm on it. I know how to do one digit integer but not 2 digit integer. Could you please help me out?

You'll need to convert the 2 digit integers [0-99] and then load to the registers.
Do your math in the registers and then convert back.