Which is correct?
- Ahmet'e kahveyi veriyorum
Or
- Kahveyi Ahmet'e veriyorum

And when I have two objects, how can I decide which is said first?

Question

Which is correct?
- Ahmet'e kahveyi veriyorum
Or
- Kahveyi Ahmet'e veriyorum

And when I have two objects, how can I decide which is said first?

Answer

It's same meaning dont be stres ;)
Nobody care about it, in Turkey. It ısn't important ;)

Answer

both are okay you just played with object 
same structure it doesn't matter u have more subjects or not 
example: 
 Kalemi ve silgiyi Ahmet'e veriyorum
 Ahmet'e kalemi ve silgiyi veriyorum.

both are okay and have same meaning. they just put at the beginning of sentence to make a stress on that.

Answer

Yine bir örnek,

Eğer Ahmet senin sevgilin ise, "Ahmet ' e kahveyi veriyorum" olabilir.

Eğer sen bir garson isen "Kahveyi Ahmet ' e veriyorum" olabilir.

Sanırım anladın :)

Answer

both  are  okay 
you  just played with object
 same structure it doesn't matter u have more subjects or not

example:  Kalemi ve silgiyi  Ahmet'e veriyorum 
 
                Ahmet'e kalemi ve silgiyi veriyorum.

both are okay and have same meaning. they just put at the beginning of sentence to make a stress on that.