There could be many ways and it depends on the speaker. However, I always reply this way
Thank you. Same here. Or
Thank you. Nice talking to you as well
This conveys that your are thankful to the person. And you too feel the same.
“How do you do” in modern American English is a formal noise made upon meeting someone. It rarely, if ever, indicates actual Interest in an answer. It can be acknowledged by repeating the same phrase back, or by saying “Very well, and you?”, to which the answer is also “Very well”. It can be abused as an opportunity to seize control of the conversation by interpreting it as one of the others, but that's rude.
“How are you” is an inquiry about the person's health, usually physical but mental/mood is included . Again it's often just a ritual, a friendly greeting, but it's not unreasonable to give a real answer to this one: “Great except for my allergies kicking up”, for example. Generally the answer should be brief, so the querant can decide whether they want to ask for more details or not. But in most cases, some version of “Fine, and you?” is the right ritual answer unless this is asked by a close friend or your doctor.
“How are you doing” is a more general inquiry about how your life is going. Depending on context and intonation, it can range from another ritual greeting (fine, and you?) to a deep expression of concern (which can be brushed off with the ritual answer or given a more meaningful response).
Exceptions abound, but those are the most common usage patterns in my dialect, anyway. (NE US)
Best Answer
Yes, when communicating via text one can say either "talk to you tomorrow" or "see you tomorrow".
Saying "see you tomorrow" means "see you online tomorrow" (even if you are not online at the same time).
It is expected that you will communicate further then, just as saying "see you tomorrow" for in-person interaction conveys the expectation that you will speak with one another, not the expectation that you will meet, stare at each other without speaking for some time, and then leave. Although that could also apply to some interactions. ;-)