2010-11: Semyon Koshelev played with Torpedo Ust-Komenogorsk's youth program in his native Kazahkstan and played for the Metallurg Novokuznetsk U16 team in the Russian Championships. In five games for Metallurg he scored 3 goals with 2 assists and was +2 with no penalties.
2011-12: Koshelev skated with Kazzink-Torpedo in the second division of the Kazakh league as a 16-year-old and played for Kazakhstan's U18 team in the 2012 Division 1B World Junior Championship. He scored 8 goals with 5 assists and was -44 with 6 penalty minutes in 36 games for Kazzink-Torpedo. Koshelev scored 2 goals with 1 assist and was +1 in five games at the D1B WJC as Kazakhstan finished second to Belarus in the six-team tournament.
2012-13: Koshelev played 47 games with Kazzinc-Torpedo in his second season in the Kazakhstan league and skated for the Kazakhstan U20 and U18 national teams. He scored 12 goals with 16 assists and was -25 with 4 penalty minutes for Kazzinc-Torpedo. Koshelev scored 1 goal with 1 assist and was -1 in five games as Kazakhstan finished second to Poland in the 2013 U20 Division 1B World Junior Championship. Koshelev led all scorers at the 2013 D1B U18 World Junior Championship as Kazakhstan finished first. In five games he scored 6 goals with 7 assists and was +7 with 10 penalty minutes.
2013-14: Koshelev moved up to Kazzinc-Torpedo's main club; skating in 47 regular season games and six playoff contests in Russia's VHL (second league). He scored 4 goals with 10 assists and was -5 with 8 penalty minutes during the regular season and was +1 with 1 goal in the VHL playoffs. Koshelev played four games with the Kazzinc-Torpedo second team in domestic play and was +2 with 1 goal and 1 assist. He was the fourth-leading scorer at the 2014 Division 1B World Championships — scoring 3 goals with 6 assists and finishing +2 with no penalties — as Kazakhstan's U20 team finished second to Italy. Koshelev shared the scoring lead with Norway's Martin Gran at the 2014 Division 1A WJC playing for Kazakhstan's U18 team. He scored 4 goals with 4 assists and was +4 in five games as Kazakhstan finished third. Koshelev was ranked 35th amongst European skaters in Central Scouting's final rankings prior to the 2014 NHL Draft.