If you're a new designer starting a logo, here's an outline you can use to begin the process. I kept this vague because the idea is for you to tailor your service to support your work but this outline will be a good beginning.
1) Explain yourself
One of the biggest fears I encounter from clients is the thought that hiring one person "locks" them into fewer choices and they might be stuck with something they don't want. It is your job to help assuage uncertainty by having a process that incorporates discussion and rationale for choices. By explaining this process you can show your capabilities and eliminate the thinking of design as a lottery or contest where the outcome *might* be positive. A process has the greatest chance of success. Design is a service, not a contest.
2) Ask questions, and be responsive
By now you have heard a fair amount about the business but it's is the time to learn more. You aren't just learning about the client's service but the industry in general: trends, competition, biggest concerns, their customers, concerns of their customer, shirt color of their customers, everything about their customers. You're trying to appeal to your client's customers - not just your client. Remember this and imagine you're working for two now (your clients and their customers). How would their customers view the work you've done?
Research is separate from asking questions because it includes what you might not hear from your client. There are things they might not know or forget to consider but your fresh perspective will help to find those things out. If there's nothing new to learn from here that's great, but be careful that you haven't simply missed something. Stay open.
4) Start loose
Now you can take your research and fulfill the promise you talked about in your outline to the client. Start working loosely. You're in the idea stage if you haven't started there yet. Don't make one idea precious. Nothing perfect, please.
5) Refine over time
Refining is where the vague takes shape and a logo comes to be. Think of a tree trunk - wide at the bottom and narrow at the top. I limit revisions because we're working toward a goal and changing elements because of too much feedback can pull us out of designing with intent and into designing to be finished. All that research is meant to guide your choices. This part of the project looks different for everyone and is a good time for your client to learn about how you work.
6) Arrive at a final, prepare files and stay helpful.
Arriving at the final logo means you tested your work at different sizes, limited color situations, and in the most difficult instances you can imagine. This can be: crammed onto the back of a t-shirt next to a dozen other logos or printed on a small one color newspaper ad. Prepare for the worst and make sure the files you send can minimize problems. Label/organize your files clearly and imagine you won't be the designer working with the logo in the future. Imagine there won't be a designer - make them easy to understand for everyone.
7) Remember your purpose a.k.a. design is a service
This is the step that exists within the rest of the steps. It is important to remember that you are not making art here (save that for later) you are providing a service. Be accessible, helpful, and open. Don't design in secret, and if you do record the process to help explain your choices.