Honestly, some of it is just practice until you have memorized the notes by playing pieces. You can also use cue cards. When you have seen 99 times that the note on the first/bottom line of the G clef staff is the E right above middle C, chances are that the hundredth time you'll have started to remember that it's the E right above middle C.
I have even read an article critiquing 'Every Good Boy Does Fine' and 'FACE' because it adds extra steps to recognizing the note, i e
1. See the note
2. Remember 'Every Good Boy Does Fine' or FACE
3. Remember where on the staff the mnemonic starts and which direction you go (ie top and down or bottom and up)
4. Count to identify which word/letter in the mnemonic the line or space in the staff corresponds to
5. Identify the letter
6. Translate the letter into a piano key
You probably shouldn't get into the habit of doing this as your crutch. When memorizing the notes directly, the process might be more like:
1. See the note
2a. Either recognize which letter it corresponds to because you have memorized it or
2b. Look it up on a cue card or chart or similar to refresh your memory
3. Translate the letter into a piano key