You can start talking to Siri just by bringing iPhone to your ear, like making a phone call. If the screen isn’t on, first press the Sleep/Wake or Home button. You’ll hear two quick beeps to indicate Siri is listening. Then start talking.
Turn on Raise to Speak: Go to Settings > General > Siri.
If Siri doesn’t respond when you bring iPhone to your ear, start with the screen facing you, so your hand rotates on the way up.