Do we need to memorise the half equations of reduction of permanganate and dichromate ions and the colour changes observed ?
If not, are there any ways that school trials or nesa can test us on them ?
The half-equations are on the data-sheet given in the HSC exam so yes you can be asked about them and you don't need to specifically memorise them as they are already on the sheet, just know where they are.
The colour changes you need to know. For example, a question that could be asked is "A student tested two isomers of C3H6O (propanal and propanone) by adding acidified permanagate to each compound. Explain the observations the student would see when conducting these tests"
The main observation would be in reference to the colour change during the oxidation.