Always be prepared! (Who said that?) Before you talk to carriers about possibly accepting you to represent their respective companies, you must do your research about them. With the technology at-hand these days, there really is no excuse for not being knowledgeable about each carrier's history, news, current published plans and at least a little bit about everything in-between. It helps you "stand-out" from the competition. You can also talk to other agents who already represent them. Among your other preparation strategies, you need to show initiative in the form of interest in them as potential business partners.
Just as importantly, don't get caught in a discussion with them without a written marketing plan. It does not have to be elaborate and some carriers will require that you complete one in their desired format. However, when you are asked if you have a written marketing plan and you have to answer "No", it puts you in an uncomfortable position and saddles you with a decided disadvantage.
A good marketing plan does not normally have to be a complete business plan, with extensive financial projections. Carriers are typically more interested in how you are going to attract and write the desired, profitable business they want. Additionally, the marketing plan does not have to be ten-plus pages in length. A few pages will usually suffice - definitely think quality vs. quantity. Start with general contact info, of course with experience backgrounds, including specialties. Then describe your agency operations in a couple of paragraphs. Next, express your strategies for acquiring the prospects you will hope to turn-into clients and carrier customers. It is important to be realistic. You should expect to be asked exactly how you are going to implement each strategy and the obstacles you may need to overcome to do so. Be as specific as possible.
There is a lot of free information available, to give you some good ideas as to what to include. I can also provide you with my thoughts, if you want them. I collect marketing strategies.
Always be prepared. It will serve you well in your efforts.
There are of course several ways to approach carriers in your efforts to get appointed. However, there is one good approach that is often overlooked.
Assuming you have developed some extent of a network of agency friends and business acquaintances, find out which ones already represent the carrier or carriers that you have determined will be best for your needs. Ask them if they will refer you, in terms of a simple call by them to their carrier rep or someone in a higher position, if feasible. Beforehand, give the potential 'fellow agent referrer' a list of your references, along with key strategies you will employ to attract the kind of profitable business that you have determined the particular carrier likes to write. You should know this based on your prior research about the carrier.