Nice to meet you.
This is short for it is nice to meet you.
Glad to meet you.
This is short for I am glad to meet you.
Generally, they mean the same thing. When meeting someone (usually, for the first time), it is a positive experience. As such, I am glad has a bit more of a personal touch, I think: you actually refer to a personal emotion of being glad, whereas it is nice is more of a general observation. The difference is subtle, though, and probably most people might not notice it. Compare for instance these two sentences:
It's nice weather today.
I like the weather today.
They roughly mean the same thing, but the second one is more personal. (And of course, if I like weather that is not usually considered nice, they may mean something different.)
Nice to see you.
Short for it is nice to see you, you use this when you meet someone you already know; as a rule of thumb, use meet when you meet someone for the first time, use see when you are acquainted with them.
Nice to found you.
This sentence is grammatical, but it means you enjoy creating someone. If you mean you are happy about finding someone, make sure you use the verb find, not the verb found.
Nice to find you here!
Glad that I found you!
These are both correct uses of find, and express happiness about having encountered someone (at this specific location).
Glad to is followed by an infinitive, so glad to found you would refer to the verb found as we see it in the foundation of a company, the Founding Fathers and the founder of a city. The man who founded a city is not the same as one that found it!
“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
There could be many ways and it depends on the speaker. However, I always reply this way
This conveys that your are thankful to the person. And you too feel the same.