The pair have both signed three-year contracts with the Ladbrokes Championship club.
Tavernier spent the second half of last season on loan with Bristol City, and never hit it off in the Premier League with Newcastle.
A bit light-weight, and always prone to lapses of concentration.
The 23-year-old right-back started his career at Newcastle and has also had loan spells with Gateshead, Carlisle, MK Dons, Sheffield Wednesday, Shrewsbury and Rotherham.
Tavernier revealed the reputation of Rangers and the chance to work with manager Mark Warburton had attracted him to Glasgow.
He told the club's official website: "Rangers are a massive club and as soon as there was interest there was no thinking about it.
"The history of the club, what they have achieved in Europe and in Scotland made it an easy decision for me.
"It is a great opportunity to play up here and the recognition that you can get playing for Rangers is huge."
He added: "As soon as I spoke to Mark he had already sold it to me, the way he wants to go and the way he wants to play fitted my football.
"Everything about it was perfect for me and the platform he has put the club on made it an easy decision."
Tavernier is looking for a hat-trick of promotions after helping Rotherham and Bristol City move up divisions in the past two seasons.
"You get different kinds of pressure with a promotion race, some lads feel it in different ways, and for me I like to play with a bit of pressure, it always makes me perform better and I enjoy it," he said.
THE TOON'S GREATEST 100 PLAYERS ... EVER!
This book is a collection of the 100 Greatest Newcastle United players of all time, as voted for by NewcastleUtd-Mad.co.uk readers and a panel of journalists who contribute to the website. The players are in an order of five-per-chapter, counting down from 100 to Newcastle United's greatest ever player (as viewed by the author), but the whole point of the book is for YOU (the reader) to decide who goes where. The list is not, by any means, definitive. As with all books of this nature, whether it be "Greatest Albums"; "Best Movies Of All Time"; or "Top 100 Pin-Ups Of The Year" ... no two people have exactly the same opinion. There will be controversy over who is on the list, and chances are, more controversy over who is not. Here are the best of the best. I have painted the picture with each player's profile. Where they end up in the league of NUFC's Greatest is up to you. Because YOU have the final word.
Otherwise ... if you want a signed copy (for a birthday, etc) for £9.95 and no postage ... you can meet me before a match and I will bring a copy along.
Get in touch: Kevnumad@fsmail.net
NEWCASTLE UNITED
Fri 10th July (8pm BST)
Gateshead FC 0 Newcastle Utd 1 (Cisse pen)
(First half): Darlow, Janmaat, Haidara, Lascelles, Williamson, Anita, Colback, Gouffran, Cisse, Sissoko, De Jong. (Second half): Woodman, Bigirimana, Satka, Good, Ferguson, Tiote, Obertan, Aarons, Perez, Armstrong, Riviere.
Tue 14th July (8pm CT)
Club Atlas (MEX) 2 Newcastle Utd 1 (Sissoko)
Krul (Darlow 46); Janmaat (Bigirimana 72), Williamson, Lascelles, Haidara (Ferguson 72); Anita, Colback; Sissoko (Obertan 72), de Jong (Perez 72), Gouffran (Aarons 72); Cisse (Armstrong 72).
Sat 18th July (8pm UTC)
Sacramento Republic FC (USA) 0 Newcastle Utd 1 (og)
Krul (Woodman 46), Janmaat, Williamson, Lascelles (Taylor 46), Haïdara, Anita, Colback, Sissoko, Obertan (Aarons 69), de Jong (Perez 69), Cissé (Wijnaldum 69).
Tue 21st July (7.30pm UTC)
Portland Timbers (USA) at Providence Park, Portland (USA)
Sun 26th July (1pm BST)
Sheffield United at Bramall Lane
Wed 29th July (7.45pm BST)
York City at Bootham Crescent
Sat 1st August (3pm BST)
Borussia Mönchengladbach (GER) at St. James’ Park