Born in the Leicestershire town of Loughborough, Moore began his career with Leicester City, where he was captain of the youth team that reached the quarter-finals of the 2010–11 FA Youth Cup.
Moore signed a one-month loan contract with Bradford City on 5 August 2011. He made his senior debut on 6 August 2011 in a 2–1 defeat away to Aldershot. Moore's announced his desire to stay with Bradford City at the end of the initial month's duration, and his loan was later extended until 31 December 2011. Moore returned to his parent club following the end of his loan deal on 31 December 2011. Moore ended his Bradford loan with 22 appearances to his name.
After returning to Leicester, Moore made his debut for them two days later, starting the game against Crystal Palace on 2 January 2012. After becoming a first-team regular at the start of the 2012–13 season, Moore was scouted by Arsenal, Manchester United, Tottenham Hotspur, Everton and Aston Villa. He signed a new three-year contract with the club in September 2012.
On 20 February 2013, Moore joined Brentford on loan until the end of the 2012–13 season. He made seven appearances that season.
Moore was recalled to Leicester on 30 March 2013, to act as cover during club captain Wes Morgan's suspension. Moore scored his first ever senior goal for Leicester on 14 September 2013, scoring the opener in an eventual 2–0 home victory over Wigan Athletic. His impressive performances early on in the 2013–14 season saw him being named the Football League Young Player of the Month for September 2013. Moore has credited club captain Wes Morgan and manager Nigel Pearson with his improved form.
In January 2014 he was the subject of transfer bids from Premier League club Fulham.
Moore made his Premier League debut in Leicester's opening day, playing the entirety of a 2–2 draw at home to Everton on 16 August 2014.
On 26 February 2015, Moore joined Brentford on loan until the end of the 2014–15 season. He went straight into the starting lineup, displacing Harlee Dean, before being dropped from the squad in favour of Dean after three "stuttering" appearances. Moore failed to appear again and was recalled to the King Power Stadium on 2 April.
He signed on loan for Bristol City on 1 September 2015, until 4 January 2016.
Moore signed a four-year contract with Reading on 20 August 2016 to seal a permanent switch from Leicester. He scored his first goal for Reading against Blackburn Rovers on 17 December 2016. The club reached the playoffs at the end of his first season with them; Moore was described as playing a "key role". Moore signed a new four-year contract with Reading on 2 August 2017.
Moore has represented England at under-17 and under-21 youth international level.
On 8 November 2012, he was called up to the England under-21 team for the first time for a friendly against Northern Ireland. He made his debut in the game, coming on as a substitute in the 83rd minute for Jack Robinson. He was recalled to the under-21 squad in November 2013. Moore helped England reach the 2015 UEFA European Under-21 Championship by scoring in the second leg of the play-offs, which England won 2–1 against Croatia. England had already won the first leg 2–1, which meant they progressed to the finals with a 4–2 aggregate win.
Moore is eligible to represent Jamaica internationally, through descent.As of 17 September 2017
Leicester CityFootball League Championship: 2013–14
Football League Young Player of the Month: September 2013