The first thing I would do, personally, is file an extension and submit payment for the amount owed. This accomplishes two important things: it gives you an extra six months to finish up your tax return, and it gets the taxes paid. This eliminates the double stress of owing the IRS and a tight deadline. And once you have that extra breathing room, you'll be able to go back to your tax return afresh and make sure all the numbers are calculated correctly and you can see if there's any deductions or tax credits you might be eligible for. You might also want to ask a tax accountant to thoroughly review your taxes to make sure your tax bill is as low as possible.
As for submitting payments, I personally use the Electronic Federal Tax Payment System, or EFTPS for short. This is a Web service operated by the Treasury Department for processing federal tax payments. Once you set up your profile and bank information, you can make payments for various tax matters, such as extension payments, estimated taxes, even paying tax balances for previous years as the result of an audit, notice or amended return. I like the ability to schedule my payments in advance, and the payments are then automatically withdrawn from my checking account.
You can also send in payments the old-fashioned way, by mailing a check with the relevant tax form. The IRS has different addresses for payments. Here's the list of addresses for estimated tax payments, and addresses for extension payments, and addresses if you use Form 1040-V. (This Form 1040-V is the payment voucher you mail in if you filed your tax return electronically.) One important tip for mailing in checks, make sure the check is written to the "United States Treasury." Some unscrupulous IRS agents have been arrested in past years for stealing checks made out to "IRS" and altering it to "MRS." or "MR. S" or "J.R.S" or other combinations that could be made out of the IRS letters.
If you cannot afford to pay in full, please file an extension before the April 15th deadline. This will greatly reduce the penalties the IRS will assess on your balance due. The IRS charges a late filing penalty, a late payment penalty, and interest on any balances owed. If you file an extension and then file your return by the extended deadline of October 15th, you will eliminate the late filing penalty of 5% per month. This can go a long way to ensuring that you don't pay more than you need to the IRS.
Also, you may want to consider setting up a monthly payment plan, called an installment agreement in tax jargon. This will help pay off your balance over time. The IRS will still charge the late payment penalty and interest, as well as a one-time user fee to set up the payment plan. You can even set up your payment plan online, using the Online Payment Agreement Application on the IRS Web site. And if you cannot afford to pay monthly, you should consider seeking advice from a licensed tax professional to evaluate other ways for resolving your tax debts.