I think a lot of who you consider rivals is dependent on who the big games are against when you first properly start supporting Alty.
However, the weird thing is I don't actually have anything against either Southport and Barrow. I enjoy it when we beat them but I don't actively dislike them. There seems like there should be another category that aren't exactly rivals but you do really sort of despise them.
For those who began watching Alty in the 1960s or 1970s,
the big derby was always against Macc (usually involving playing them on Boxing Day and New Year's Day).
Owing to Macc's failure to qualify for the Alliance Premier League for the 1979/80 season, the green slime of Vics emerged to become our bête noire (notably assisted by the heinous Jeff Forshaw).
Macc didn't attain promotion into the then GM Vauxhall Conference until the 1987/88 season.
As regards teams from afar who you loathe, well, any Alty supporter who watched the Robins during the 1970s would surely add Boston United to that particular list of infamy.