fertig
finished
Describing a state in which an activity, process, or work is completed or ended.
Das Essen ist fertig, kommt bitte an den Tisch!
The food is ready; please come to the table!

finished / ready
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finished
Describing a state in which an activity, process, or work is completed or ended.
Das Essen ist fertig, kommt bitte an den Tisch!
The food is ready; please come to the table!

To express what you have finished, you use the preposition 'mit' followed by the dative. This structure is very common.
Bist du schon mit den Hausaufgaben fertig?
Are you already finished with your homework?
When 'fertig' (ready) means that you are ready for something, it can stand alone or be used with 'für' + accusative or 'zum' + nominalized verb.
Ich bin fertig, wir können gehen.
I'm ready; we can go.
In colloquial language, 'fertig sein' (to be exhausted) means that you are extremely tired or worn out. You can intensify this with adverbs like 'total' (totally) or 'völlig' (completely).
Nach der langen Reise war ich einfach nur noch fertig.
After the long trip, I was simply completely exhausted.
The expression 'jemanden fertigmachen' has a very negative meaning. It means to wear someone down psychologically, criticize them harshly, or bully them.
Sein Chef hat ihn vor allen Kollegen fertiggemacht.
His boss tore him down in front of all his colleagues.
through / by
'durch' and 'fertig' can both express that a task or piece of work is finished. 'fertig' is the normal and neutral word for this. You can say: 'Ich bin fertig' (I’m finished), 'die Arbeit ist fertig' (the work is finished), 'das Essen ist fertig' (the food is ready). 'durch' is used in 'durch sein' (to be done) more colloquially. It often sounds as if you have worked through a task or gotten something behind you. In many situations, 'fertig' is more natural, especially in neutral language. 'Ich bin durch' (I’m done) fits more in conversation when you are finished with a list, a text, or a piece of work.

Paul ist mit dem Abendessen fertig und deckt den Tisch.
Paul is finished with dinner and is setting the table.
Nach drei Stunden Lernen bin ich mit dem Kapitel endlich durch.
After three hours of studying, I’m finally done with the chapter.