You give everyone a large standard deduction -- $14,500 sounds about right, because that's what a minimum wage earner working 2000 hours per year grosses. Also have a $3100 deduction for each child -- 20% of what an adult gets.
Have a few other deductions for extraordinary expenses, such as college tuition, medical bills, charitable contributions etc... but not so many that it becomes a clusterfuck.
Apply the tax system to all income -- labor, capital gains, corporate profits, etc -- and then set the rate at whatever makes it revenue neutral.
So to do some back of the envelope math -- in 2015 the US government took in $3.34 trillion in revenue.
GDP is around $16.77 trillion.
The population is around 319 million, 20% of whom are children. So using the figures above, total personal deductions would come to about $4 trillion. Let's round it up to $4.5 just to be safe.
This creates a tax base of about $12.27 trillion. So to collect $3.34 in revenue, the tax rate would be about 27%.
How about the Feds just start spending less also? How about that? 
A person struggling to get by on min wage pays zero.
The average worker earning $25k would pay about $2800 in tax, for an effective rate of only 11%.
A person making six figures would owe $23k, for an effective rate of 23%. And so on and so forth.
Keep in mind this replaces all other taxes. No more payroll taxes that assrape the poor and middle class, no more loopholes for the rich and well-connected, etc. It's fair across the board, and still progressive without getting into brackets and all that bullshit.
I have a Masters in Taxation. My first day the professor says, "you know why taxation is so complicated? So you have to pay someone to do it for you." If it was simple, the government couldn't debate ridiculous tax code, and we wouldn't need accountants to do our business and personal taxes. 
Stronghold - How about the Feds just start spending less also? How about that?
Absolutely agree, but spending is kind of a separate discussion.
Magic_Mike - I have a Masters in Taxation. My first day the professor says, "you know why taxation is so complicated? So you have to pay someone to do it for you." If it was simple, the government couldn't debate ridiculous tax code, and we wouldn't need accountants to do our business and personal taxes.
Very true -- politicians and their cronies use it to game the system and increase their own power & bottom line.
Just because it's shitty now doesn't mean it has to be, though.
Information - You need to look up the definition of GDP.
Circular flow model
Magic_Mike - I have a Masters in Taxation. My first day the professor says, "you know why taxation is so complicated? So you have to pay someone to do it for you." If it was simple, the government couldn't debate ridiculous tax code, and we wouldn't need accountants to do our business and personal taxes.
Me too. and I absolutely agree with you. The reason a flat tax or simplified tax will never happen is the amount of special interests involved in keeping it the way it is.
...as is the problem with most of the laws and issues in this country. they big ones are complicated because lots of different groups feel lots of different ways.
What if you pay for your own kids instead?
attjack - What if you pay for your own kids instead?
Don't really understand the question.
The point of a standard deduction is to allow people to avoid tax on basic living expenses like groceries, rent, fuel, etc -- without having to itemize. These things are income to businesses so they will pay tax on their side.
Children are people who have expenses of their own, therefore they should be counted as part of the deduction process.
Nothing will change.
If the purpose of taxation was simply to fund the government, they yes taxes should and WOULD be simple.
But taxes have evolved into a tool for social engineering. The government isn't going to give up that power.