Ideally, you should factory reset the device, so you're starting afresh.
You haven't mentioned what device you have or what version of Android you're running, but usually the setting to factory restore a device is found within the main SETTINGS > BACKUP & RESET. Use the Factory reset option to erase all personal data from the device and start afresh. Once the device is rebooted, you'll be prompted to enter your Google account credentials and link your phone and all apps you download from the Play Store linked to your Google account.
Be warned, factory resetting your device will erase all personal data on the device. If there's any content, you want to retain, copy it over to your PC or to a cloud service like Google Drive or Dropbox (using your account, not the previous owner's).