What are the steps studied to develop a calculation note for the compressed air tank according to the ASME standard? What is the best design software that makes this calculation easier?

The place to start is to have in front of you a copy of the ASME VIII standard and to read through the standard thoroughly. Before considering any software you may apply to the process, a complete understanding of the code requirements to design, build, test and have certified your completed compressed air tank.

My experience of designing and building pressure vessels is limited to a few vacuum tanks. in my part of the world the standard that applies to the design, construction, testing and certification of these pieces of equipment is AS/NZS 1200. By starting the project by reading the applicable standard makes the whole process relatively straight forward. All of the required calculations are laid out in the standard, specifications for materials and welding and what is required to test and certify the finished product, you start at the beginning and work your way step by step towards the finished product.

When it comes to using finite element analysis software to do the calculations, if you have the software available to use, you may make use of this tool. But as part of your question is asking about software options I would guess you have limited experience applying this tool in the real world. I found following the calculations in AS/NZS 1200 to be not a difficult process, hardly worth the bother and expense of purchasing a product, then learning how to use this software correctly. As with all software tools we make use of in engineering design, finite element analysis software does not do the thinking, we the engineer do this part. to get the right answer requires telling the software the right inputs, you need to fully understand the loading and stress involved in pressure vessels. This you learn from the technical education you have received along with a good understanding of the design code you are applying.

At the end of the project the drawings, design calculations, manual or done using software, along with testing data gets bundled up and sent to the certifying engineer, If you have done every thing correctly will issue a certificate that allows your product to go into use.

The notes you will place on your drawing will reference the code that the tank is designed, built and tested to. The certifying engineer will inform you if it needs any changes.

If you are working in a job where designing air receives and other types of tanks and pressure vessels may become a large part of your work look to build up a library of useful textbooks I have attached a scan of the cover and contents page from a text helpful in this sort of design.

I should start by saying I'm oil field and we do things to a different code, but I have designed multiple tools to +10,000 psi internal and external (hydraulic).

To my knowledge, the idiom "it's boiler Plate" comes from the codes that covered the design/documentation from top to bottom. I would first say, read the codes and follow them. As for software to help, I've never found calculations that weren't straightforward enough to write out in excel, maybe with some VBA to cover iterative calcs.

FEA: Man, I have a hard time telling someone that they can just run an FEA and call it good. While programs can be useful, the first thing I have to say is Garbage in = Garbage Out. I work in Solidworks and had to completely change the material library to match the real world materials to even get in the right ball park. I would suggest starting with a very simple model (Thick walled pressure cylinder of Unit Length) and try modeling it, and check it vs Von Meises Calcs by hand (less the 3% difference is acceptable). There are lots of good tutorials online if you need a refresher on the calc. After that you can go more in depth. If this is human life dependent, i still wouldn't trust it.
Note: FEA wont match code. Design Codes are inherently conservative. So even if you run an FEA, I bet the code will say the vessel can hold way less the math comes out to be. I did a calc about water hammering caused by the time it takes to close a valve, and it took me 6 academic papers to find out that a safety factor of 5 is just thrown in for good measure so that you don't blow up your pipes.

For this particular case. I would also look into scuba tanks for a design reference. They have a really good hydrostatic testing procedure. I actually have my scuba instructing buddy test my air tanks for me when I build them from scratch.

Thick Walled Pressure Vessel Refresher:
https://courses.washington.edu/me354a/Thick%20Walled%20Cylinders.pdf