Buy or borrow an audio interface to match the outputs of your reel-to-reel tape recorder. If your tape recorder uses RCA outputs you can find interfaces as low as $50, but if it uses XLR three-prong outputs, it'll cost $100 or less. (See Reference 1.)
Download and install Audacity, a free music recording software program for Mac and PC. (See Reference 2.)
Connect the tape recorder to the audio interface and then plug the interface into the USB or FireWire port (whichever one it uses). Open Audacity and go to “Preferences.” Select your newly installed audio interface under "Audio I/O" and under "File Formats" choose "WAV 16-bit PCM" and click okay to exit preferences.
Click the red recording button and then click the pause button on Audacity. Under the microphone input, select monitor input.
Thread the tape on your recorder and hit play. Set the output balances on the tape recorder, or select automatic if such a switch exists. Set the input levels on Audacity and when all looks well with the software, rewind the tape to the beginning.
Hit the pause button on Audacity and start recording. Push play on the tape recorder and monitor the recording through headphones from either the computer or your tape recorder. When finished, press the brown stop button on Audacity.
Click "File" in Audacity and select "Export as MP3." Your WAV will be converted and ready for your portable MP3 player.