A research career cannot begin without completing your doctoral studies. Once you have obtained your PhD diploma, you will naturally be in a position to look for a job — one where your skills are valued. Since such jobs aren't easy to find, it would be frustrating to waste the experience accumulated during your doctoral school. This is why the concept of a postdoc emerged.
The postdoc is not another stage of your studies. It is your first real job as an independent researcher.
Most often, postdoc positions are available where a project director wants to build a research team and needs people with proven experience. If you have just graduated, all you need to do is look for these types of offers. The disadvantage is that you will be integrated into a team working in the direction set by the project, and the data you manage belongs to the project — so you often won't work on the exact topics you dreamed of. The big advantage is that you have a team, usually well-established leadership, secured funding for investigations and your salary, and success is almost guaranteed.
Another way to start is to write and win your own postdoc project. In Romania, UEFISCDI offers this through the Human Resources programme, under the subprogramme Start in Research — Postdoctoral Research Projects. This time, you have the opportunity to capitalize on your own ideas by proposing a project exactly as you wish. All you need to do is convince a mentor with the appropriate expertise (ideally, you should have collaborated previously). If your project proposal is accepted for funding, it will be conducted in an institution — university or research institute — but according to the work plan you devised and exclusively from the financial resources you have acquired.
I am convinced that any research institution in Romania will welcome you. This is the big advantage of this type of project — the opportunity to actually create your own job, defining your path for the first two years of your career. However, during this time, it would be wise to already be working on your next project, aiming for the Young Teams competition. If you win that one too, not only will your CV look excellent — you will find that the institution where you implemented these projects will do everything possible to keep you.