The main reasons a supporter may start to hate another team:
4 Personal. A supporter may have personal reasons for hating another team, not shared by any others of the same club. For example when visiting another teams ground, having a bad experience like a poor quality pie leading to a uncomfortable journey home or an on-going football team based conversation over months or years between two work about their respective football teams.
3 Frequency of a certain fixture. If two teams regularly meet over several seasons a rivalry can develop. This could be common amongst teams who routinely finish in the top half of the Premier League e.g. Manchester City, Manchester United, Liverpool, Arsenal, Tottenham Hotspur and Chelsea. Unlike in other Leagues or Divisions their is no promotion meaning repeat fixtures reoccur season after season, potentially compounded with cup competitions.
2 Historical hatred. A hating of another team may be due to historical factors. For example when Leeds United popular in the 1970s they would have attracted haters due to their success or other fan's jealousy which has become engrained over time despite a fall in Leeds United's success. Indeed supporters of certain clubs perpetuate such hatred to new generations who were not born when the the hating started.
1 Local rivalry. Supporters of local teams are most likely to hate each other because of the banter which is inescapable in school, workplace or pub. The teams in question may regularly or rarely play each other but the "community effect" never goes away. For example Celtic and Rangers in Glasgow, or the close proximity of the City of Newcastle and City of Sunderland and their respective teams; Newcastle United and Sunderland. Similar local rivalry could be extended to international fixtures like between England and Scotland.
