Are there any animals a zoo MUST have to be considered good?
Yes - healthy, normally behaving animals whose needs are met as extensively as possible and who the zoo can efficiently take care of.
For example, if a zoo didn’t have elephants, would it be considered bad?
No. Elephants are not a guarantee for quality.
If not bad, are there any species a zoo must have to be your favorite?
Take another look at my first answer. And it's not just the animals. In many previous discussions, the complexity of evaluating a zoo has been debated over and over again. The quality of a (modern) zoo is not measured by the amount of popular species it keeps. Otherwise, many atrocious roadside zoos would be "good" zoos. The quality of care (and that alone includes hundreds of aspects), the quality of the staff (and the way they are treated, trained and paid), the efficiency of business management and finances, the quality of visitor services and amenities (nobody likes filthy restrooms), the participation in conservation, research, education, national & international networking, sustainability, animal welfare, community work, inclusion of minorities etc.,the general "embedment" of the zoo in its environmental and social neighborhood, its online visibility in the social media etc. etc. - there are many, many factors to account for. And there are of course the individual personal preferences of zoo nerds: some only like the typical ABC species of the charismatic megafauna and can recite their individual lineages and names in almost religious joy. Some prefer zoos with more "unusual" or rare species, while others are more interested in zoo history / architecture, behavioral enrichment, animal training, technologies, zoo landscape, dining options, entertainment features, ample cheap parking lots, connections to public transport (as quite a lot of zoo fans don't seem to have a driving license or a car), or just put emphasis on how easily they can take good pictures of animals.
All in all, the overall impression, how the zoo manages to fulfill all these requirements and your individual understanding and knowledge of all of this can contribute to you deciding whether it's a good zoo or not. I can go to a zoo that has none of the species I'm fond of and still enjoy my visit. Or I can go to one that has all my favourites and still disappoints.
While there are certain general things we can agree on, a lot is based on your individual criteria.