For Spotlight Search, you need to swipe down from the approximate center of any Home Screen, not down from the top of the screen. If you swipe down from too high on the screen, you'll bring down Notification Center. If you still can't access Spotlight Search using the described method, try a device reset:
Device Reset (won't affect settings/data/music/apps/etc)
1. Press and hold (& continue to hold) BOTH the Sleep/Wake button & the Home button.
2. Continue to hold BOTH (ignoring any other messages that may show) until you see the Apple logo on the screen.
3. Release BOTH buttons when you see the Apple logo and allow the device to boot normally.