Visit your state's Division of Corporations on the web. Most states list the incorporation requirements there. The requirements will usually include submitting a form listing your chosen name, address, and business of the company, the names and addresses of the initial board of directors if required, your chosen number of allowable shares (called 'Number of Authorized Shares' in most state statutes), and some language relating to the number of classes of shareholders and their rights, and other governance issues. For LLCs, some states may also require you to list the names and addresses of the LLC's initial 'Members' (the equivalent of 'Shareholders' for corporations). A filing fee somewhere between $50 and $250 is common.
After incorporating, many states require LLCs to issue public notice via a minimum number of local newspapers. Most newspapers charge for the public notice service, and public notice listings costs in large counties can reach several thousand dollars. Most states do not require public notice for corporations, making the total cost of incorporating a corporation significantly lower than incorporating a LLC in most cases. You'll likely also be required by law to create bylaws or an operating agreement for the company and document annual or other regular meetings and keep those documents on record.
Your own name and home or business address will often be required since you are the 'Incorporator' of the company. Incorporation filings are part of the public record in most states, and any information listed in your filings will be accessible by anyone for a fee or free. There are incorporation services called 'Registered Agents' which will list themselves as the incorporator of your company in exchange for a fee. Completely concealing your identity when making government filings is impossible and sometimes illegal in most jurisdictions. Hiring a registered agent will put one more step between your name and the public though, making it more difficult for interested parties to ascertain ownership of your company and any assets you transfer to it.
Corporations are taxed as independent entities without recourse to their shareholders' or officers' assets. LLCs are generally not taxed, but their members are taxed based on their share of assets of the company. This is called 'Flow Through Taxation' in most state and federal statutes because the LLC's assets will 'flow through' to you as an owner of the company and you will be taxed on any taxable income as if it were your own personal income. Most states also require regular fees and/or minimum taxes for all companies regardless of revenue.
Incorporating a company is a simple process and designed so that members of the public can complete the process themselves without the aid of a lawyer or other adviser. If you set aside the time to read and understand your state's incorporation law and rules, you can comfortably incorporate your company or companies yourself. If speed is important in your situation and you are willing to pay your lawyer's incorporation flat fee or billable hours, you can call or visit your lawyer for a short consultation and s/he should be able to complete the incorporation within 24 hours to one week. A longer completion period would be highly unusual.
Best of luck.