How exactly does one use the verb 欢迎?
Like in the sentence, 欢迎您到北京来.
Does it require 到 when saying welcome to? What is the purpose of 来 in the sentence?
欢迎您"到"(arrive)北京"来"(come). = 欢迎您"来到"(to come to and arrive in)北京. - Welcome you to come to (and arrive in) Bejing.
The above sentence is the more formal way of greeting. Less formally, you can say:
欢迎您到北京 - Welcome you to arrive in Bejing.
欢迎您来北京 - Welcome you to come to Bejing.
歡迎 often means "welcome/greet (a guest etc.)" or a related meaning "someone/something has a good reception"
Example of the first use in a standard way is saying "歡迎您到北京來!(we) welcome you to beijing!" note the formal nin, this is a fairly formal vocab in this form. Imagine going to greet a guest for pick up at the airport and make them feel welcome, sign and entourage and all: this encapsulates the mood of this word.
The second use can be casual or formal, often in reviews and so on: an example is saying "這場電子遊戲深受公眾歡迎 this video game has been well received/is popular(with people in general or a specific group in context etc.)"
Though you asked for how it is used as a verb explicitly and others here have already provided good examples of this, there are ways it can be used as an adverb or adjective as well. Most of these can be found by looking up 歡迎 in the Pleco dictionary directly, with some example sentences below:
Visitors are welcome. 歡迎參觀。
The streets were lined with welcoming crowds. 大街兩旁擠滿了歡迎的人群。
Literary works of this kind are very popular. 這樣的文學作品很受群衆歡迎。