There is not a step-by-step estimation for design work. Rules of thumbs can be used, but vary so greatly between projects because of the amount of work needed / not needed by each owner.
A full-service architect will charge based upon those services needed. So what you need to decide and discuss with your architect is what services do you need.
We often get folks who found a floor plan online and "just need it stamped by an architect". Once we show them that the plan will have concerns on their lot (windows under ground, design is for a mountain home not their beach property, etc). they soon realize that while the plan they found has features they like, it doesn't work for their life style, location, lot, etc.
So the first step is Programming. This is the process where the architect will discuss with you want you want and need. A young family planning for children, vs an established couple looking toward retirement will have different considerations. It is not uncommon during this phase to realize that many decisions need to still be made (She wants a single level, he wants a 2-story on a basement!). This process also has allowed our clients to find houses for sale that fit their needs perfectly, because they had them written out and understood. This is typically an hourly rate of $60 to $200, depending on the location. This also will typically include code and zone research (HOA requirements, etc).
The next step is schematic design. Taking the needs from Programming, the architect will layout the rooms and travel areas to provide a custom design that includes all the needs, and as many of the 'wants' as they can, while looking at the site conditions and offerings (views, driveway locations, etc). This is often combined with the programming stage during contract negotiations; you will get a list of what you need, what you want, and what obstacles / requirements you face and a few sketches / renderings showing potential layouts, and maybe some elevations.
Typically an architect will hold a retainer (a down payment). This amount will be a percentage of the estimated value of the project. So 5 to 12%. This retainer is held, and paid toward the final invoice. It can be negotiated, so if you only want to go to Schematic design, it might be lower than if you want full construction documents. If you continue to use the architect, they will request a second retainer to increase the amount held - the idea is the architect does not hold risk that you will change your mind (decide not to build) and then not pay for the designs up to the point they have been developed.
After schematic design is Design Development. This is where the details are hashed out and actual sizes come into play. Instead of "this area could be the living room" it becomes a 12 x15 living room, etc. Products begin being selected and if you have a builder you want to use, they will be brought in for comment and suggestions for saving cost, making the construction easier, etc. At this stage, you will be able to get a more trustworthy budget in place to know the actual cost of the building. Hard decisions are made that affect the design and costs.
Finally you get to Construction Documents. These are the plans that the builder and permit office will use to build your custom home. All design decisions should be made and the look of the house established. Residential design is typically a bit more expensive than commercial design (it has a lot more options, less Code requirements driving decisions, and more "hand holding"), so a total fee of 8 to 15% of the construction cost is possible. Before you get a calculator and start thinking of that as a large amount, keep in mind that a good architect will providing savings well over 15% the cost by selecting proven materials, designing for minimal maintenance, designing for added benefit to the user, and providing expert guidance during construction. Good design saves money.
There is also construction management; the architect will work with the builder to ensure the building matches the plans and the intent, as well as handle any in-field issues that pop up during construction. This also covers making sure the project is on time and using the correct materials, products, etc. This is optional.
There are also bank options, the architect can prepare back documents to assist you with securing the loan, etc. And there is bidding and negotiation; the architect can help you will getting bids and negotiating with the contractor. Each of these is an additional fee, usually paid in hours with the architect giving you a set number of hours. These are optional, and not needed on every project. They are great peace-of-mind offerings though that let you know you are getting what you paid for and don't have to make decisions or worry about if your home is being constructed correctly, etc.