На даче мы ели арбуз, а потом жарили сосиски на огне.

This is an idiomatic expression in Russian.

• дача = a summer house, country cottage, or garden property
• на даче literally looks like on the dacha, but in real usage it means at the dacha / at the country house

Russian often uses на with certain places where English uses at or in. На даче is just the normal phrase you learn as a set expression.

Compare:

• на даче = at the dacha
• на работе = at work
• на кухне = in the kitchen

So this is not something you translate word-for-word too strictly; it is the standard Russian way to say it.

даче is in the prepositional case.

The preposition на can require the prepositional case when it means location:

• дача → dictionary form
• на даче = at the dacha

This is a regular feminine noun pattern:

• дача
• о даче
• на даче

So the ending -е shows that the noun is in the prepositional case after на.

ели is the past tense plural form of the verb есть = to eat.

Because the subject is мы = we, the verb has to be plural in the past tense:

• я ел / ела = I ate
• он ел = he ate
• она ела = she ate
• мы ели = we ate

This verb is a little irregular, so it is worth memorizing:

• infinitive: есть
• past plural: ели

арбуз is the direct object, so it is in the accusative case. But for many inanimate masculine nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: арбуз
• accusative: арбуз

That is why the form does not change.

Compare:

• Я вижу стол. = I see a table.
• Мы ели арбуз. = We ate watermelon.

Both стол and арбуз are inanimate masculine nouns, so the accusative looks the same as the basic form.

сосиски is also a direct object, so it is in the accusative case. But сосиски is plural, and for inanimate plural nouns, the accusative usually looks the same as the nominative.

So:

• nominative plural: сосиски
• accusative plural: сосиски

That is why the form stays the same here.

жарили is the past plural form of жарить, which means to fry, roast, grill.

Here it means something like:

• we grilled sausages
• we cooked sausages over the fire

Because the subject is мы, the past tense is plural:

• мы жарили

This verb is in the imperfective aspect, which is natural here because the sentence is describing what they were doing as part of a sequence of events, not strongly emphasizing the completed result.

Russian often uses the imperfective past to describe actions in a narrative, especially when simply telling what happened as activities:

• мы ели арбуз
• потом жарили сосиски

This gives a natural storytelling feel: we were eating watermelon, and then we grilled sausages.

If you used perfective verbs, the focus would shift more toward completion or one-time completed actions. That may also be possible in some contexts, but the imperfective here sounds very normal for describing a relaxed sequence of summer activities.

Both а and и can connect clauses, but they do not feel exactly the same.

• и = and
• а often means and / while / whereas / then, with a slight sense of transition or contrast

In this sentence, а потом feels like:

• and then
• and after that

It smoothly moves the story from one action to the next:

• first we ate watermelon,
• then we grilled sausages.

So а here is not a strong contrast like but. It is more of a natural shift to the next event.

на огне literally means on the fire, but in natural English it usually means:

• over the fire
• on an open fire
• over the flames

So жарили сосиски на огне means they were cooking the sausages using open fire, like at a campfire or outdoor grill.

Because after на in the meaning of location or position, Russian uses the prepositional case.

The noun is:

• огонь = fire

In the prepositional case:

• на огне = on the fire / over the fire

This is not the most predictable-looking change, so it is often best to memorize it as a form:

• огонь
• на огне

Russian often can leave out subject pronouns when they are understood from the verb form.

So both are possible:

• Мы ели арбуз, а потом жарили сосиски на огне.
• Ели арбуз, а потом жарили сосиски на огне.

The version with мы sounds slightly more explicit and clear. It may also help set the scene or emphasize we as the people doing these actions.

So мы is not strictly necessary, but it is perfectly natural.

This is a very normal, neutral word order.

• На даче = sets the scene first
• мы = subject
• ели арбуз = first action
• а потом жарили сосиски на огне = second action

Russian word order is flexible, but this version sounds natural because it presents:

1. the place,
2. the people,
3. what they did,
4. what happened next.

You could move things around, but this sentence already sounds like standard, natural narration.

Russian does not have articles like English a and the.

So:

• арбуз can mean a watermelon or the watermelon
• сосиски can mean sausages, some sausages, or the sausages

The exact meaning depends on context. In this sentence, English can translate it naturally as:

• We ate watermelon, and then grilled sausages over the fire or
• We ate a watermelon, and then grilled the sausages over the fire

Russian leaves that unstated unless the context makes it specific.