self-employment tax is about 15% of total income which based on the $25,000 you specified is by itself $3,750. in addition, there would be regular income taxes on the balance but you may be entitled to the QBI deduction which could chop about $4000 off your taxable income if you qualify.

state estimated tax payments must be made separately to the state - not the IRS. 

Sorry none of your thinking is right.

You pay self employment tax only to federal not state.  Self Employment tax (Scheduled SE) is automatically generated if a person has $400 or more of net profit from self-employment.  You pay 15.3% SE tax on 92.35% of your Net Profit (If it is greater than $400).  The 15.3% self employed SE Tax is to pay both the employer part and employee part of Social Security and Medicare.  So you get social security credit for it when you retire.  

You pay both regular income tax and self employment tax on your Net Profit.  The 12,500 Standard Deduction is only deducted from your total income to calculate the regular income tax, not the SE Tax. 

Do you have any expenses on the self employment income?  Expenses go on Schedule C also and can reduce your Profit or even give you a loss which you can deduct against your other income.

The quarterly estimated Payments are only for federal to cover the se tax and any regular income tax.   You pay state estimates separately to your state.

So if you don't have any expenses to write off or your Net Profit is 25,000 you will pay regular income tax on the 12,500 taxable income (25,0000-12,500 Standard Deduction).

AND pay about 15.3% self employment tax on the full 25,000 which is about $3,532.

So I guess your $3,750 should cover it but remember it is just for Federal.   You have to figure state separately.

Oh the regular income tax on 12,500 taxable income for Single is $1,306 plus the $3,532 SE Tax = $4,838 total tax.  But you may qualify for some credits like the EIC Earned Income Credit which can reduce the regular income tax but not the SE Tax.