In an IRA, if I buy a stock and sell it a year or two later but then buy something else without withdrawing any money, is there any paperwork I would need to do when filing taxes during the year when I sold it? Or would I only need to worry about the paperwork when it's time to withdraw the money?
There is no paperwork to be filed with the IRS regarding stock transactions within an IRA. All gains and losses on stocks or mutual funds or bonds held within an IRA stay in the IRA and are totally irrelevant as far as IRS reporting is concerned. You don't need to save the details because they will never be reported outside the IRA.
You don't need to worry all that much paperwork when you withdraw the money because your IRA custodian will send you Form 1099-R telling you how much cash you have withdrawn and what (if any) part of this cash has been sent to the IRS as withheld income tax. If you have requested that the money be sent to you abroad, there is a mandatory 30% tax withholding; but for US residents, you can choose how much to have withheld. In either case, you simply file Form 8606 along with Form 1040 reporting the withdrawal and how much tax was withheld from it. Your custodian gave you stock shares that you are currently holding inside the IRA instead of cash? Well, technically the IRA custodian sold the shares at the closing price of the shares in the day of the distribution, gave you than amount a cash, and you promptly bought the same number of shares at the closing price, all without any brokerage fees or bid/ask spreads to worry about. The transaction will still be reported as cash paid out on Form 1099-R.
If your State taxes IRA distributions, you might need to file State Forms 1040-ES to pay state income taxes on a timely basis. Many IRA custodians don't offer withholding of State taxes at all.