You may need a COVID Certificate to prove that you have received both of your vaccination in order to travel internationally.
Policies may change in the future, allowing travel with a certificate to certain countries without the need for testing.
Entry requirements vary across different countries. You may be able to enter some countries having provided a range of test results. You should check the entry and return requirement at https://www.gov.uk/foreign-travel-advice/ or https://www.dfa.ie/travel/travel-advice.