John Brown, 25, from Motherwell, admitted carrying out the offences at two separate games at Ibrox stadium.
He targeted Cowdenbeath's Kudus Oyenuga as he celebrated a goal on 28 March, and disabled Falkirk fans on 3 October.
Brown was also given a community payback order and told to carry out 70 hours unpaid work.
Glasgow Sheriff Court heard that Brown "made monkey gestures using his arms and directed them towards the Cowdenbeath football player" as he celebrated a goal.
Police then approached Brown at the stadium and told him he was being arrested.
Brown committed the second offence after Falkirk scored and equalised at the game in October.
The court heard that he reacted by shouting towards the Falkirk away fans and gestured towards the disabled section.
Following sentencing, Stephen Ferguson, football liaison prosecutor for the west of Scotland, said: "Brown's behaviour was utterly reprehensible and choosing to vent this kind of hateful abuse at football matches is no excuse at all.
"There is absolutely no place in modern Scotland for individuals who commit crimes motivated by prejudice towards a person's race or disability.
"Police and prosecutors take such offences very seriously and will continue to do all we can to bring those who persist in committing them to justice."
A Rangers fan who made monkey gestures to a black player and abused fans with disabilities has been given a 12-month football banning order.