The standard way to express the future tense is -(으)ㄹ 것이다. 것 corresponds to a set plan or fact, and -ㄹ is the verb form implying something yet to happen. This phrase is used for both plain future (i.e. expected to occur) and one's intention.
-(으)ㄹ 것이다 is the base form, and can change depending on the formality and politeness one wants.
For example, 나는 유학을 갈 것이다 (I will be studying abroad) can have different endings, of which the 거예요 you mentioned is the contracted form of one.
1) 나는 유학을 갈 것입니다 / 겁니다. (것입니다 contracts to 겁니다)
2) 나는 유학을 갈 것이다 / 거다. (것이다 -> 거다)
3) 나는 유학을 갈 것이에요 / 거예요. (것이에요 -> 거예요)
4) 나는 유학을 갈 것이야 / 거야. (것이야 -> 거야)
(In each of them, 것 becomes 거 and the ending form adapts to this new vowel-ending syllable.)
The four sentence endings (-ㅂ니다, -다, -아/어요, 아/어) are called the plain endings, meaning they state a fact without a special connotation other than the formality and politeness.
-(으)ㄹ게 (and its polite form -(으)ㄹ게요), on the other hand, is one of the special sentence endings, which have added connotations. In -(으)ㄹ게's case, it makes a statement of intention with a promise to the listener, like "I will ... (for you)" - it is often used to volunteer something that helps or otherwise affects the other person.
Example: 청소는 내가 할게. 너는 쉬어. (I will do the cleaning. You relax.)
So -(으)ㄹ게 is not really a simple future statement but a phrase intended for special cases.