The 2030 Agenda with 17 Sustainable Development Goals (SDGs) is a challenge for all countries in the world. Their implementation may turn out to be a compromise or the creation of effective interactions that dynamize sustainable development. To achieve the SDGs, it is essential to understand how they interact with each other. It seems that in the times of the climate and health crisis caused by the COVID-19 pandemic, caring for the environment and ensuring a healthy life and promoting well-being at all ages is the basis for environmental, economic and social sustainable development. The aim of the study is to compare the degree of implementation of the goals of sustainable development in the scope of goal 13 “Climate action” and goal 3 “Good health and well-being” in the EU countries. In addition, we analyze how trade-offs and synergies between these goals have developed. Data from the Eurostat database were used to achieve the goal. The study used the method of multivariate comparative analysis—linear ordering of objects. The technique for order preference by similarity to an ideal solution (TOPSIS) method was used to measure the studied phenomenon. The results indicate a different degree of implementation of the sustainable development goals related to climate change and the improvement of health and social well-being. Only a few countries have synergy in achieving these goals, most of them compromise, manifesting themselves in improving one goal over another. In the group of analyzed EU countries, a simultaneous deterioration in the effectiveness of achieving both objectives were also noted. Our research also shows that energy policy is an important attribute in improving the achievement of these goals. The conducted analysis fills the gap in the research on the implementation of selected sustainable development goals and their interactions. It contributes to the discussion on increasing the links between them, in particular with regard to emerging compromises. This research can provide a basis for re-prioritizing and intensifying the actions where individual EU countries are lagging most behind.