What is most important for smart phone speed?

You need to upgrade once security patches are no longer released for your phone’s OS.

Beyond that your paying a premium for new features which may or may not be worth it to you. I usually wait until the device I’m using can no longer run the latest OS.

“Response for searching information”?

If you mean a web search, yes that’s almost surely going to be about your network lag, not anything else.

It depends on what you are using it for at the time. Can you be more specific?