"içeyim" is the imperative form of the verb içmek for the first singular subject "I".
Let's see the imperatives of this verb for all subjects in order;
içeyim - let me drink
iç - drink!
içsin - let him drink
içelim - let us drink
için(iz) - you, drink! (it is more polite when you add -iz and works for both "you sing. and plural")
içsinler - let them drink
içmek istiyorum: I want to drink
You can say either of them when someone asks you "what you would like to drink".
for example; you are in a restaurant and the waiter asked you;
Q: What would you like to drink? Ne içmek istersiniz/isterdiniz?
A1: I would like to have (get) a cola please. Ben bir kola alayım, lütfen (almak: to take, to get) (alayım can be read as alıyım, and içeyim--içiyim; we tend to make it ı after a,ı,o,u and i after e,i,ö,ü, a from al and i from iç)
A2: I want (to drink) a cola. Ben bir kola (içmek) istiyorum.
I would use the first option, it sounds more polite.
But when this is not the case, thus you are not asked to tell what you would like to have, when you talk about what you will do or what is your plan, it is like telling something you've already decided, you made your plan already but you just let people to hear it politely. It is slightly different than saying "I want".
- Ne zaman buluşuyoruz? When will me meet up?
- Ben önce eve gideyim (can be read as gidiyim) ve biraz dinleneyim (can be read as dinleniyim), sonra seni ararım. First let me go home and take some rest, then I call you.
Hope this helps. Please let me know if you still have questions.