ALREADY is usually found in mid-position (before normal verbs, after special verbs: modals and auxiliaries) but we can put it at the end for emphasis.  YET always goes at the end
 
- I'm already tired
- She has arrived already!
- I'm not tired yet
- She hasn't arrived yet

ALREADY is used in affirmative sentences and YET in negative sentences

- I have already finished
- I haven't finished yet

ALREADY / YET in questions

We can use both in questions, but the meaning is a bit different. YET simply asks if something has happened or we still have to wait. ALREADY knows that something has happened, it simply expresses surprise because it happened sooner than expected. If we put ALREADY at the end, we are emphasizing our surprise.

- Has the doctor arrived yet?  - No, not yet, but you can wait for him, he will arrive in a few minutes.
- Your father is here already?! I thought the train arrived at 10:30 and it's only 9:45!

Compare:
- have you finished yet?  (come on, I'm waiting!)
- have you already finished?  (wow, you're so fast!) -vs- have you finished already!? (oh wow, I can't believe it!!)

for ADVANCED learners  We also use ALREADY when we suppose (but don't know for sure) that something has happened. In this case there is no surprise, simply a guess, and it always goes in mid-position, never at the end.

- Have you met Doctor Sullivan yet?  (I don't know, I'm asking)
- Have you already met Doctor Sullivan?  (probably I don't know but I think you have)

Note for Spanish Speakers

 
EQUIVALENCIAS INGLESAS AL ESPAÑOL "YA"
 
En las afirmativas usamos el "already" en mid-position. En las negativas usamos "anymore" en posición final, y en interrogativas usamos el "yet" en posición final o el "already" en mid-position si mostramos sorpresa:
 
(+) I have already finished = Ya he terminado
(-) I don't want it any more = Ya no lo quiero  (también en preguntas: ¿ya no vives aquí? = You don't live here any more?
(?) Has he arrived yet? = ¿Ya ha llegado?   /   Is he already here? = ¿Pero ya llegó?, ¡qué pronto!