The Liverpool legend's appointment as Gers manager has generated much optimism among the club's fans after going through three managers during two seasons of third-place finishes in the Scottish Premiership.
Tavernier, who was signed from Wigan Athletic by Mark Warburton in 2015, has praised his new manager's high standards and also revealed how he continues to be starstruck by the former England international.
"Most of the time it still feels strange he is my manager. When he strikes a ball with so much ease, stuff like that is a little bit surreal," Tavernier told RangersTV.
"I know he is the gaffer now and you do get those little moments when you find yourself going, 'That's Steven Gerrard over there'.
"[Gerrard and his coaches] have raised the standard higher which has always needed to be done because it has dropped over the years.
"Straight away I have noticed the standard and where he wants it and the boys have responded really well. Even from doing small passing drills and our little boxes, the level has already gone up."
Rangers are currently in Spain for pre-season training.