На даче мы поставим мангал у забора и будем жарить овощи.

Because Russian normally says на даче to mean at the dacha / at the country house property. This is an idiomatic expression.

• в usually means in / inside
• на can mean on, but it is also used for some places and areas idiomatically

So на даче does not literally mean on the dacha. It means at the dacha, as a location where people stay, relax, garden, cook, etc.

Даче is prepositional singular.

The base form is:

• дача = dacha, country house

After на when it means location, Russian uses the prepositional case:

• на даче = at the dacha

So:

• nominative: дача
• prepositional: даче

Мангал is a charcoal grill / brazier, especially the kind commonly used outdoors for шашлык and other grilled food.

It is not exactly the same as a big American backyard barbecue grill in every context, but grill or brazier is usually the best translation.

In this sentence, поставим мангал means we’ll set up / place the grill.

Поставим is the perfective future of поставить. It presents the action as a single completed action:

• поставим мангал = we will set up / put the grill in place

That makes sense here because setting up the grill is a one-time action with a clear result.

If you said будем ставить, it would sound more like we will be in the process of placing it, which is less natural here unless you specifically wanted to focus on the process.

This is an aspect choice.

• будем жарить = future imperfective
• пожарим = future perfective

Будем жарить овощи focuses on the activity/process of grilling vegetables. It suggests we’ll be grilling vegetables as something we do there.

Пожарим овощи would focus more on the vegetables being grilled as a completed result: we’ll grill the vegetables.

So the sentence contrasts nicely:

• first, a completed action: поставим мангал
• then, an ongoing activity: будем жарить овощи

Забора is genitive singular.

The dictionary form is:

• забор = fence

After the preposition у meaning by / near / next to, Russian uses the genitive case:

• у забора = by the fence / near the fence

So:

• nominative: забор
• genitive: забора

Овощи is accusative plural.

The singular is:

• овощ = vegetable

The plural is:

• nominative plural: овощи
• accusative plural: овощи

Because овощи is inanimate, the accusative plural looks the same as the nominative plural.

So in:

• будем жарить овощи

овощи is the direct object, so it is accusative.

Yes, it could be omitted.

Russian often leaves out subject pronouns when the verb ending already shows who is doing the action:

• На даче поставим мангал... is possible

But мы is still perfectly natural. It may be used for:

• clarity
• emphasis
• contrast with someone else
• a slightly more explicit, conversational style

So both are possible, but мы is not wrong or unusual.

Russian word order is more flexible than English word order.

This sentence starts with the setting:

• На даче = at the dacha

Then it gives the subject:

• мы = we

Then the actions and details:

• поставим мангал
• у забора
• и будем жарить овощи

This order sounds natural because it sets the scene first: At the dacha, we’ll set up a grill by the fence and grill vegetables.

Other word orders are possible, but they can change the emphasis.

All three can mean something like by / near the fence, but they are not identical in feel.

• у забора = by the fence, right next to it
• около забора = near the fence
• возле забора = near / beside the fence

У often gives a strong sense of being right by something, so it works very naturally here if the grill is going to stand close to the fence.

Russian has no articles like a and the.

So у забора could be translated as:

• by the fence
• by a fence

The exact meaning depends on context. In many real situations, English would naturally say the fence, because the listener can identify which fence is meant.

The same applies to мангал and овощи in many contexts: Russian leaves article choices to context, while English usually has to choose one.