A sole proprietor or single-member LLC are the same, assuming you aren't electing to be treated as a corporation. You are the company. All your income is yours personally and so just added to your taxable income. However, this is done through a Schedule C on your tax forms.
You could run everything thing though your personal bank. It's just how you account for it that matters. Income and business expenses are logged as such in your books and recorded in Schedule C. There are no accounts to prepare unlike a corporation.
However, your bank may not like this, depending on what business you have and the amount of transactions you generate. So you could either set up a credit card, and just put all business expenses through that, or set up a business bank account and credit card for keep it completely separate. But in the end, both are your accounts, the business doesn't really exist as a tax entity. It is just convenient.
Income that comes in, less any bona fida business expenses, gives your self-employed net profit. This is going to be taxed with at 15.3% for social security and medicare and then added to your personal taxable income. Most tax filing software have SE versions that will handle this for you.
You aren't going to get taxed twice, because there are no business taxes for your situation, it's all personal income.
Other things you need to consider:
Register with your state for their tax, as applicable, e.g. make sure you are paying income tax or B&O tax
Get your business license with your city if necessary and find out if they charge taxes.
Make sure your professional indemnity insurance covers enough.
Wouldn't harm to get a CPA that you can make sure everything is set up and ask questions later.
Check where your health-insurance comes from, marketplace?, and the deductions that you can take.
Set up a self-employed 401k/SEP IRA if you plan to make contributions.
Make sure every client you engage / work you do is covered by a contract (This will be needed for your insurance but you SHOULD do this anyway)
You will be paying estimated taxes in April, June, September and January. Check you liability from last year to understand your minimum this year, otherwise a very general rule of thumb, pay 30% of your income.