Translation:Let's go eat!
Most common in English.
"Let's eat" means you already have the food in front of you.
"Let's go (to) eat!" means you have to go where the food is. Also "to" is not kept. With or without "to" meaning is same.
People use language and languages change by that use. Use will eventually cause the guardians to amend their rule books. But, somebody has to guard the towers and prevent total linguistic chaos.
Suppose there's a bar we know, but we didn't realized it served food. Now someone tells us they do serve food and it's great. We might say..."so...let's go to eat." This illustrates a point of transformational grammar. Namely that a sentence feature that usually gets deleted at a late stage in generation may not get deleted when we need/want it to carry stress. Here one stresses "to eat" versus "to drink."
"let's go to eat" still sounds weird here. In this context you'd say " (oh they serve food too?) Well lets go and eat" I think "go to eat" is so outdated that it no longer sounds correct.
Exception is if you say "i go to eat but" to show that you we're about to do an action but something else happens, however, you'd never use "let's" with that context.
Actually it's a tricky one in terms of formality
자 ending is a part of a "formal impolite" speech level. But contrary to this, it is not used very often in any kind formal speaking, in fact is used as a kind of panmal supplement (panmal or 반말 is the lowest, least formal speech level).
So basically, it kinda is informal in everyday life. But from technical viewpoint it is considered as a formal ending. No idea why.
Sometimes the correction will not allow the slightest deviation from the hidden vocabulary prompt, while other times it insists on using something entirely unprompted, with the excuse that one has to be idiomatic in English. The student has no way of knowing if the "teacher" is a literalist stickler or a free translation philosopher. This is bad pedagogy. I wrote "in order to," knowing it was unidiomatic but yet the prompt used it. If additional rules are to be given, make them explicit, perhaps in parentheses. Otherwise, being scored wrong for being cooperative is inducement to quit.