-(y)Alİm is only used for the optative/hortative, so only for let us...!, shall we...?. In Spanish (I see you're learning this language too) you would form this modality with the subjunctive present, like in hagámoslo (let's do it), levantémonos (let's get up), veamos (let's see).
Btw the suffix for the singular optative (or hortative?) is -(y)Ayİm and depicts an intention or desire of the speaker, but I think that will come later in the course.
To say "let me pay the bill" I believe you have to use bırakmak - to let. So it should be "bırak bana ödemek" or "ödemek bana bırak". I'm just guessing, would be nice if some native could correct me :)