I learned the treble clef first by a long way.
I learned it by the mnemonic FACE.
When I eventually came to learn the bass clef I simply used the mnemonic ACEG.
After quite a short while it just became natural.
I didn't ever do any calculations of any kind.
I knew that middle C was shared.
Later I connected the lower ledger lines on the treble clef with the upper notes on the bass clef.
This happened automatically.
My recommendation Don't do any kind of calculation.
Just learn the lines and spaces as you did when you learned the treble clef.
music.stackexchange.com/users/21436/chasly - reinstate Monica
Don’t read by adding semitones, this is a wrong way.
You better start reading the bass clef by studying and practicing a piece like this: Mind that the upper ledger lines of the lower system are the corresponding lower lines of the upper system and vice versa.
Study the link
In the comment of guidot.
The real cognitive insight is not transposing the names in the system.
What you must realize is that the ledger lines c,a,f,d are identical with the lines of the bass system.
When understanding this point you are able to read bass, tenor and alt clefs: you don’T have to transpose 3 semitones, you just move the system up and down.
music.stackexchange.com/users/56219/Albrecht Hügli
"My question is: what is a 'right' cognitive process to read sheet music fluently?
I wonder how one does not confuse reading notes in different clefs."
I am not sure there is a 'right' process, or method.
Some of the methods we are taught as children are, in my opinion, very flawed and create more distraction than actual learning.
The best method is repetition.
You don't want to have to go through the math every time you change clefs but the more practice
you get the sooner the training wheels come off.
I think Albrecht Hugli provided a good answer, though it seems to be down voted.
For me what works is the realization that everything is relative.
All you really need to do is identify one note on the clef and pair it with your instrument.
A good example is middle C.
Then all the other lines and spaces just follow the alphabet (A -> G, repeat).
It really can't get easier than that.
Sight reading and reading for a specific instrument are different skills.
On the guitar for example you need to not only read the note but decide which version of it to play since some notes can be found in 3 or 4 places.
You can practice reading the notes by letter name out loud independently of finding them on your instrument.
That is a good way to eliminate the crutch of "Every Good Boy Does Fine", or "Add three semitones", etc.
There are books out there that recommend just this exercise.
Once I started doing that after a short while all other methods just dropped out of my mind
and I just ID
the letter immediately regardless of clef.
music.stackexchange.com/users/50691/ggcg
It's not really wrong but it's better to learn the bass (and alto and tenor a bit later) clef as a thing in itself.
Reading and playing piano and organ music helps.
The point is to think of the Grand Staff as the object of interest, (for keyboards) with a bass clef, middle C ledger line, then treble clef.
Later you can learn the C-clefs.
It just takes practice.
Lots of practice.
music.stackexchange.com/users/23838/ttw
When you read English (or whatever your native language is, assuming it's alphabetic), do you read the letters individually and sound them out to form words?
Of course not; long ago you achieved reading fluency.
Same with music - with enough practice you just look at the score and automatically play what's in front of you.
It doesn't matter if one staff is in treble and the other in bass, or if both staffs are in bass, or if the top staff is in bass and the bottom in treble for some hand-crossing passage, or if the composer decided to write in 3 or 4 staffs to make the voicing more clear.
Just as there are debates (see phonics vs whole language) about the best way for children to learn reading English, there are debates about how people should learn to read music, but, just as for English, it probably doesn't matter that much in the long run.
If you start playing chamber music with cellists or violists, you learn to fluently read tenor and alto clef
so you can follow what they're doing.
Probably you can depend on your tenor clef experience for B-flat instruments (though the few measures Brahms wrote for clarinet in A-sharp are quite annoying).
music.stackexchange.com/users/21176/Alexander Woo
There will be better answers but for in short
I believe your method is okay.
Because now you are counting them but
soon after you will start to memorize them one by one.
In short time it looks efficient.
Also to memorize them you can draw a staff and write all notes on it once a day.
It may help you to memorize them so that you will stop counting.
Also you can try this website
music.stackexchange.com/users/66587/Nabla
I have a question about practising reading in different clefs.  I know the treble clef well. When I read a note in the bass clef,  I mentally add 3 semitones to it, so I get the note name in the treble clef.However, I'm suspecting that this method will hinder my sightreading speed: I'm not reading the note instantly.  My question is: what is a 'right' cognitive process to read sheet music fluently? I wonder how one does not confuse reading notes in different clefs.
Sheet music reading and cognitive process?
music.stackexchange.com
<sight-reading>
music.stackexchange.com/questions/104435
music.stackexchange.com/users/71726/markhollis
The best method is repetition.  The more practice you get the easier it gets over time. The best method is repetition. 
The best method is repetition. All you really need to do is identify one note on the clef and pair it with your instrument.  What works is the realization that everything is relative. It doesn't matter if one staff is in treble and the other in bass, or if both staffs are in bass. Just as there are debates about the best way for children to learn reading English, there are also debates about how people should learn to read music.