After you file your return, you will want to keep a copy of the return for your own records. Tax returns have a way of being useful even after you file, such as documenting your income for a loan. From a strictly tax perspective, you'll want to keep your returns
for at least three years. That's how long the IRS has to audit your return. Along with the actual tax forms themselves, keep any documents concerning your income, deductions, and various tax credits.
I waste enough paper at the office printing out returns for all my clients, so I'm always looking for ways to store more records electronically. If your software provides a copy of the return in a PDF format, you could save that to your computer. I would go one extra step and put a copy on a USB drive and put that in a safe place like a safe deposit box. This way you'll have backup copies in at least a couple of places. This year I also scanned all my income documents and tax deductible receipts into a PDF file as well, so hopefully my filing is a little more environmentally-friendly this year than it has been in the past.
If you purchased desktop software this year, you should make backup copies of your tax data file as well, and keep that in a safe place along with the original installation files. This way if you need to make corrections, you'll still have the software and will won't have to recreate your return from scratch. Also, the tax data file will be useful next year, as programs can import data from last year's file to prepopulate data for the next year.