Tavernier, 23, completed a move from Wigan Athletic to Ibrox a week ago, and enjoyed a stunning competitive debut for the club on Saturday afternoon.
The right back brought Rangers level at 1-1 against Hibernian with a superb free kick, setting Mark Warburton's men on their way to a huge 6-2 win in the Petrofac Training Cup.
Fans were mightily impressed with Tavernier's performance, and they can expect more from a player who has been starved of first team football in recent years.
Tavernier was something of a forgotten man at Newcastle United, and whilst he joined Wigan last summer, he failed to make an impact there either.
Now, speaking to the Daily Record, Tavernier admitted that he's glad to be playing for a big club again having left Newcastle - but added that playing at Ibrox is even more intense than St. James' Park.
"I started off at Newcastle and, while some of my other clubs have been great, I feel as though I’m back at a proper big club now," said Tavernier. "I’d say it’s even more intense at Ibrox than it is at St James’ Park. You get more feeling from the fans here. The Newcastle supporters are very passionate but here it is something else. It’s great to be back playing at another massive club."
"When I was young at Newcastle they were a club with loads of money to spend on international players. I was only 18 or 19 when Danny Simpson and James Perch were ahead of me. But then Debuchy came in after having a great World Cup – so you can hardly complain about not playing."
"But I feel as if this journey at Rangers is just beginning and I want stability now in my career. I want to have a season when I’m just playing football every week. That’s why I couldn’t turn down a move to a huge club like this," he added.
Tavernier showed what he can do against Hibs, displaying superb pace and the ability to make a real impact in the final third - both with his goal and with his link up play.
After disappointing displays from Richard Foster and Sebastien Faure at right back last season, Rangers now have a dependable right back for years to come in Tavernier.
With experience of playing for a big club before, Tavernier won't be overawed by playing at Ibrox - and he has everything to become a star for Rangers under Warburton.