Taylor made his first team debut for Kilmarnock on 14 May 2016, starting at left back in a Scottish Premiership game against Dundee United.

[5] He then made his second start in the second leg of the Scottish Premiership Play-off Final, which Kilmarnock won 4–0 against Falkirk to win the tie 4–1 on aggregate.

[12] On 18 August 2020, Taylor scored his first goal for Celtic in a UEFA Champions League qualifying match against KR Reykjavík in a 6-0 victory.

In a dramatic match which seen both sides draw each other 3-3 after extra-time, Celtic won on penalties and lifted their 12th successive domestic honour as part of an unprecedented quadruple treble.

[30] Taylor was named in the full Scotland squad for UEFA Euro 2020 qualifying fixtures in June 2019;[31] he remained on the bench in the home game against Cyprus at Hampden Park.

After captain Andy Robertson picked up an injury, Taylor debuted at left-back against the world's top-ranked team, Belgium on 11 June, and played the full match in Brussels as Scotland lost 3–0.

[35] His younger brother Ally Taylor is also a footballer,[3] and made his first team debut for Kilmarnock in February 2020 while Greg was playing for their opponents Celtic.