While фашист and бандит should translate in English respectively to 'fascist' and 'bandit', I have the feeling that their real meaning is much stronger in Russian than in English, indeed "you're a fascist" and "you're a bandit" sounds relatively low in English and these expressions are rather soft or even can be funny in English ( and in other languages ) while they are - I feel - much stronger in Russian ( or is that only Ukrainian ?). Is that impression correct or is that an illusion?
Фашист in colloquial Russian means primarily a German Nazi (or, through metonimy, any supporter of Nazi or Nazi-like ideology).
The term was widely used by the Soviet media in 1930's, and the German troops which invaded Soviet Union were labeled "German Fascist invaders" (немецко-фашистские захватчики) right away.
In stricter sense of word, it would mean quite the same as English "fascist".
Because of the cultural connotations, the term фашист would be deemed a very strong expression by majority of Russian speakers born or raised in the former USSR. I personally can't imagine a situation where it could be used jokingly.
Бандит means a "gangster", a member of an armed organized crime group. It's a legal term as well. Contrary to its English counterpart ("bandit"), it does not have positive connotations.
More correct translation of "bandit" to Russian would be разбойник. For instance, "good" guys like Robin Hood and Hong Gildong are usually called разбойник, but Al Capone or Tony Soprano would rather be бандит.
It's not quite a strong term, though. For instance, a mother could say он такой бандит! about her child, meaning the child does not behave.