You've made me spend a few minutes paying attention to how my mouth works. It's been very interesting, because I've never really done it before. I'm going to give you a little insight to the feeling of each sound, but just the basic ones that each letter makes most often. There are actually many more sounds than there are letters.
The vowels are special and set apart because they're made by vocalizing and just changing the shape of the tongue, cheeks, and lips, but keeping them far enough that they don't produce any hissing, buzzing, or popping sounds. A, as in apple, is an open throat, open lips, tongue pulled back, cheeks tense. E as in cheese is cheeks pulled back (like a smile) and jaws about halfway open. I, as in vice, actually moves from O (as in odd) to E (as in cheese). If you say the short O and then a long E you've said the long I. Sorry for switching between short and long pronunciations. I'm choosing the ones I think work best. O as in goat puts your lips in a circle, tongue pushed a bit forward. U as in fun is done exactly like A(apple) but with cheeks relaxed.
The letters L, R, W, Y, Z, J, V are a lot like the vowels, and you could technically spell a word phonetically using these and leaving out some vowels (frvr for fervor, mrdrr for murderer). The difference is, I think, that there's a little more mouth manipulation in this list than the vowels. L is a vocal sound with the tip of the tongue against the back of the teeth. R is like O(goat), but with the tongue pushed forward to take up more space. W is like O(goat) but with the lips tighter. Y is like E(cheese) but with the throat closed a little more. Z is a vocalization with the tongue almost nestled into the back of the upper incisors, with just enough space to make a buzz. The sides of the tongue are gently between the molars during Z. J is like Z, but the teeth are together and lips are pooched. V is just like U(justice) but with the lower lip against the upper incisors.
F and S are like the previous group, but with no vocalization from the throat. Just air. F is identical to V, and S is identical to Z, minus the voice.
B, D and G create sounds by closing the sinuses, forcing a gentle air pressure into the mouth, and then opening a certain part of the mouth a split second later. These are done with voice. With B, the air bursts from the lips, with D the air bursts from the tongue, which forms a pocket around the roof of the mouth, with G the air bursts from the throat.
P,T,K, and C are like the previous group except without voice. P is like B, T is like D, K is like G, and C sometimes makes an identical sound to K.
N and M are done by closing off a part of the mouth and vocalizing through the nose. With the N the tongue is placed like D or T. The M is with the lips together.
The H is just exhaling with the throat tensed.
People say the Q is a kwa sound, but since it's always followed by U, it's effectively identical to K and some C.
X is the K sound moving to the S sound.
As simple as all of this sounds (ha ha) a lot of sounds actually, when analyzed, are completely different sounds depending on what letter follows, because the mouth is always preparing for the next sound, and this changes the shape of the mouth. We don't hear the difference, and a lot of people wouldn't believe it was there. The mouth works more quickly and automatically than the fingers of a skilled typist, but that's a good illustration of what it feels like to talk. Mindless typing.
I don't know if you had this kind of description in mind, but to be honest this taught me a lot and once I started I couldn't stop. At one point I deleted it all and started over because I realized how the sounds could be grouped.