Requirements differ based on country and zoo.
In Germany, a zoo keepers needs to have "Ausbildung" (vocational secondary education), basically, after high school (Matura), you get accepted at a zoo to do a 3-year program that is a combination of school lessons and practical work in a zoo. After 3 years you pass exams and you are a qualified zoo keeper that can start looking for jobs. Free places for Ausbildung in Germany are highly competitive. Zoos don´t hire people without Ausbildung certificate for keeper jobs.
In the Czech republic, we have 2 vocational schools where you can study to be a zookeeper. You start after you get basic education (15 years old) and it´s a 4-years program with Maturita exam at the end, education is free but you pay ca 40eur/month for boarding (usually 2 people per room so normal student accomodation). One school has an agreement with Prague zoo and its students do their internship there. However, way less than 50% of keepers in Czech zoos have any education related to biology, animal keeping or veterinary. When zoos hire people, they will see it as an advantage, but you can get hired without it. Turnover is high, people are starry-eyed only to realise the job is not filled with cuddling animals the whole day and pay is abysmal. Also, economic cycle is important, CZ zoos have problem to find employees and will accept almost anybody when unemployment is low (I have seen positions advertised for many weeks during years 2017-2019). Zoos post open jobs on their website and you need to send a CV, that is all, any relevant experience and related education will just increase your chance to be hired.