Learning to draw entity-relationship diagram for a scenario involving Offices, Employees and Projects
Good. The essentials are there. A first rough list of entities is to take the nouns of the story - office, employee, project, clients, assignment, skill. Each table has a unique, sequential, meaningless key. Columns are given descriptive names. Here are the suggestions by entity. 

office> The manager is an employee and should have a separate foreign key relationship with manager_employee_id. There is a many-to-many with employee. Avoid M-to-M relationships. In this case, an office has many employees. But an employee has one home office. I suggest naming fields uniquely with a common suffix describing the domain, office_address, instead of address. 

employee> The job_id should be a foreign key to job instead of title. The office_id should be a foreign key to office instead of location. The column names should be hire_date instead of the start_date to avoid confusion with the project start_date. Use employee_address instead of address based on the domain suffix rule. 

project> The project manager is a separate foreign key to employee pm_employee_id. What is the business difference between estimated_cost and quoted_cost? Make sure there is a foreign key field to match every relationship line. Client_id should be the foreign key. Office_id should be a foreign key. 

assignment> Good, this is an intersecting table between project and employee. I suggest renaming the date to assigned_date to keep the suffix consistent. Notice that the hours_worked are in this table with the assigned_date. The assigned_date happens once. But the hours worked happen daily. 

daily_assignment> Maybe there should be a separate child table called daily_assignment with its own primary key, a foreign key to assignment, and the worked_date and hours_worked as fields. This tracks each item of data uniquely at its own level. 

client> 
I suggest renaming the address to client_address. 

There are ways to make it more complicated. But I think this model works well. It should give you flexibility to make business changes, yet still be simple enough to code.