One doesn't start out to become a lobbyist in Washington, DC, or a state capital. Instead, one becomes an expert on something, and then becoming a lobbyist is a natural outgrowth of that expertise.
If one wishes to be formally recognized as a lobbyist, then there are various registration requirements and annual updates. But one can also lobby informally without registering, as long as certain lines are not crossed. The difference is that formal lobbyists can conduct various fundraising activities legally and can have other "access" to lawmakers, in exchange for registration and public disclosure of their activities. But anyone can "lobby" for a cause by contacting elected officials by phone, email, or in-person meeting.
An example might illustrate. Let's say that you are very interested in environmental issues. You come to Washington, DC, to work at for a nonprofit group that is dedicated to saving the rainforests. Your job is to be a researcher who assesses the rate of deforestation in South American rainforests, and to work with various governmental and non-governmental groups to minimize deforestation. You do that work for five years, and you enjoy it. You learn a lot, you make great contacts in the environmental community, and you build a working familiarity with business and political issues that often conflict with environmental goals. Then, you decide that you would like to try to solve the problem more directly, rather than being a researcher. You decide to become a lobbyist. You have expertise that you can bring to the task - whether it's directly on rainforest issues, sustainable development in South America, or environmental issues in general. You can now work with any number of environmental organizations as a lobbyist. Your job will adjust from being a researcher to using work by other researchers to bring to light the issues that are of concern to your organization. You will write advocacy papers and blogs; you will meet with other activist organizations; you will contact the media (or be accessible when they wish to contact you); and you will meet with political and business leaders to try to get them to understand the issues. All of these activities are part of being a lobbyist. The activity that draws the most attention is the last - visiting in person with elected officials and their staffs, in order to explain your views and to present evidence. That's part of the job, surely, but it's only the tip of the iceberg. Your job as a lobbyist is really the culmination of a lot of ground-laying effort to develop expertise and effective arguments. As your credibility grows, you can present that evidence to decisionmakers and (sometimes) they will even seek your input when they are writing laws.
Lobbying is a hard job, but it's very personally rewarding if you have a cause that you believe in, and if you enjoy personal meetings to advocate your viewpoint.