Gelmek- You know it is the infinitive form.
I can say that "Geleyim" is a kind of "wish", "advice" or a kind of "demand" or "obligation" form of the verb... We can understand it clearly in the text if it is a "wish, advice, demand, offer, obligation"
For example: I am ill and one of my friend offers me as "Sana geleyim, ilaç getireyim." Infact this offer has a secret question... "Do you want to come next you and bring medicine by me?"
Another one: We saw a very beautiful and cheap "something" (car, house etc.) and your wife advices you and demands from you as "Bu evi alalım." And this meaning has a bit "Let's buy this house, as you see it's so nice and cheap, may be we can't find a house like this in future, this is a kind of chance for us..."
koşmak / koşalım... (obligation)
Ex: We are walking in a mountain path and we see a wild animal, and the group leader shout as "Hızlıca koşun / koşalım" ... In here it is a kind of obligation ... And also it is a kind of offer, demand also... But you know what can be happen if you don't obey his advice...