Here is a typical question which I have been asked many times while giving public lectures in various places. While I know one of the paths like Diploma, Masters and PhD but sometimes this is not so obvious for that eager young person who thinks to go on that path. So what that would be?
You start out getting a Bachelor of Science in a related field. This could be physics, astronomy, mathematics, or possibly chemistry. Depending on which country you are planning to go to grad school in, specializing at this stage may not be as important as in later stages. However, note that in the UK, for example, it is almost unheard of for a student without a Bachelors in physics (almost always with a minor in astrophysics) to gain a place in grad school for astronomy.
After doing the BSc, you would go on to get a Masters degree in astronomy, which would set the stage for a PhD in astronomy. The PhD (and to some extent, the MSc) will have you specialize in a specific area in astronomy, such as star formation, planetary studies, or cosmology.
After doing the PhD, you would get a post-doc at a university or research institute. This is typically a three-year job where you do research in your chosen area. Most astronomers do two or three post-docs.
After this, you could become a professor or research associate at a university or an in-house research astronomer at an observatory. Universities or observatories are typically the only places to be a "professional astronomer", and depending on the position, would give you time to do your own research in conjunction with other duties like teaching.
It really depends on what you consider a "professional Astronomer" but typically you need a Ph.D. so the typical path is a Bachelor's degree (I recommend in Physics) then a Masters and Ph.D in Astronomy.
It is possible (at least it was) to get postions at some Planetaria and as a telescope operator with only a Masters. This could be considered a "professional Astronomer" postion and doesn't require the full Ph.D.
However, most places that are looking to hire an astronomer are looking for the doctoral degree.
This can vary widely depending on several factors including country in which you hope to practice as a professional, country in which you study, what you intend to do as a professional (teach vs. research) among other things. I will answer assuming one intends to study and practice in the US.
Typically, an astronomer begins with a Bachelors of Science in Astronomy or a related field (math, physics, etc.), followed by a PhD in Astronomy (this generally takes about 5-7 years). After obtaining a PhD in Astronomy, one could be called a professional astronomer, however, this is often not the end of the journey. Most astronomers will seek appointments after receiving their doctorate (of about 2-3 years in length, referred to as 'post-doc' positions). Usually after 2 such positions, an Astronomer will seek a permanent position. These include positions at NASA facilities among others, but many will seek positions at universities. The most coveted positions are those with tenure.
After getting a PhD, there are many options depending on the destination, but the above is a common track for obtaining a position at a research institution. Those aiming to teach rather than research may seek fewer post-doc positions.
The positions posted on the AAS Job Register are typical of those in a "career in astronomy," and indeed an advanced degree is a common theme, but not a universal one.
There is enough background knowledge needed for many of these jobs that picking it up by other means would be challenging, and convincing a potential employer that you've done so without a degree in physics, astronomy, or astrophysics, even more challenging (and probably only possible at all if you have a PhD in a similar discipline and already have a significant publication record in astronomy).
There are, however, many other jobs that need to get done at observatories and as part of astronomy projects, ranging from draftsman to project manager, and the requirements for those jobs do not differ all that much from the requirements for those jobs in other fields. These sorts of positions sometimes show up on the AAS Job register as well, but are usually advertised elsewhere.