'Welcome' in this use is a shorter, more casual, more personal way of saying "Welcome to (place)."
"Welcome to my home."
"Welcome to the restaurant."
"Welcome to New York."
In the first example, or if you are talking with someone familiar to you, you can reply by saying "Thank you for inviting me" or "Thank you for having me." You can also just smile and say "Thanks."
In the second example, you can simply say "Thanks" or "Thank you." If you are talking to a waiter or a host, and you want to be more outgoing, you might reply "Thank you. What do you recommend?" or "Thanks. What is today's special?"
In the third example, like the second example, when you are not familiar with the person, a simple "Thanks" will suffice.
In any case, you don't need to say thank you. Generally, you can just continue with the conversation or go about doing your own thing.