There are a number of variables here which will influence price, difficulty and time.
First is location. Some locations are more difficult to build in than others. San Francisco is notoriously difficult to build new homes in because of planning rules, protection of existing views, and a host of other things. It may also involve demolition of an existing structure. There are other places in the Bay Area which are also difficult and sometimes doing a partial demolition will be easier than a complete demolition. In any of these cases you would want to start with an Architect who has experience in the location that you are building in. They will be able to navigate through planning approvals and advise you about what sorts of constraints apply.
If you are not in an existing built area, then the issue becomes things like access and utilities. Sewage, water, electricity etc. become concerns. Again, an Architect who is familiar with these things will be valuable.
After the constraints of the site are known, design can start. Depending on the complexity, an Architect or a Design-Builder will have to get involved. They will work with you to produce a design for the building and start to work with contractors to get an estimate of the construction costs. They will also arrange for a structural or civil engineer to do the engineering necessary. Once you are OK with the design they will produce a set of drawings which are used to get a permit. This may be a two step process and need both planning approval (does the building meet planning and zoning requirements? Are there any community objections?) and then building department approval (does the design meet building and life safety codes). Usually at the time the permit set is done you would have selected a contractor and the contractor would deal with obtaining the building permit.
After that, a permit is issued and a contractor will muster their forces to execute the work. The Architect at this stage continues to participate in what is frequently called CA, in which they act as the owners agent to review the work being done to make sure it matches plans, answer questions and revise designs due to issues or changes, Some people hire a construction manager to oversee the construction and coordinate between the Architect, Engineer, Contractor etc.
An architect usually charges a percentage (7-10%) of the total construction cost to perform this job. It can vary based on negotiation. Often costs for a smaller project are higher proportionately.
Construction costs vary widely as well. In the Bay Area, I’d guess that $200 per square foot is the typical construction cost, but it can go up from there to whatever you want to spend.
How easy it is also depends on where you are, what the building is like, how good your architect and contractor are and how much money you want to spend. It is harder to do it cheaply. If you have a pile of money, a lot of your problems can be borne by others.
Time for the whole process is also variable. Some projects can take more than a year just getting through planning. Extensive site work or need for utilities can extend the duration. Complexity of the house, size of the house, available resource for the contractor, number of changes you make, weather, and other factors can all delay the project. A year would be a reasonable duration, but like cost, it can go up from there.
Finding a good Architect and Contractor who have experience in the location you are building is important. They will know how to negotiate the processes and will avoid issues with planning and permitting.