V létě chceme jet k moři, protože naše dcera se učí plavat.

Czech does not have articles like English the and a/an.

So:

• v létě can mean in summer or in the summer
• k moři can mean to the sea or, more naturally here, to the seaside
• naše dcera is already specific because our daughter is definite by meaning

Czech usually shows definiteness through context, word order, or words like ten / ta / to when extra emphasis is needed.

Because the preposition v here requires the locative case when talking about time: in summer.

The noun is:

• léto = summer

Its locative singular form is:

• v létě = in summer

So this is a case change, not a different word.

Czech often leaves out subject pronouns when they are obvious from the verb ending.

• chceme already means we want

So my is unnecessary unless you want emphasis or contrast:

• My chceme jet k moři, ale oni chtějí do hor. = We want to go to the seaside, but they want to go to the mountains.

This is very normal in Czech.

Because jet means to go by vehicle/transport, while jít means to go on foot.

• jet = go by car, train, bus, plane, etc.
• jít = walk, go on foot

For a summer trip to the seaside, jet is the natural choice:

• chceme jet k moři = we want to go to the seaside

If you were talking about habitual travel, you might use jezdit instead:

• V létě jezdíme k moři. = We go to the seaside in summer / We regularly go to the seaside in summer.

The preposition k means to, toward, up to, and it takes the dative case.

The noun is:

• moře = sea

Its dative singular form is:

• moři

So:

• k moři = to the sea / to the seaside

In this sentence, it usually means to the seaside as a holiday destination, not literally right up to the water’s edge.

Because those mean different things.

• k moři = to the seaside / to the sea as a destination
• do moře = into the sea
• na moře = out onto the sea, onto the open sea

So:

• Jet k moři = go to the seaside for a holiday
• Skočit do moře = jump into the sea
• Vyplout na moře = sail out to sea

That is why k moři is the normal phrase here.

Because protože introduces a subordinate clause, and in Czech that clause is separated by a comma.

So:

• V létě chceme jet k moři, protože naše dcera se učí plavat.

This comma is required in standard Czech punctuation.

Because náš (our) must agree with the noun in gender, number, and case.

• dcera = daughter, feminine singular nominative

So the correct form is:

• naše dcera = our daughter

Very roughly:

• náš = masculine singular
• naše = feminine singular here

This agreement is a basic feature of Czech adjectives and possessive words.

The basic verb is učit se, which means to learn or to be learning.

So:

• učí se plavat = is learning to swim

The word se is part of the verb učit se. Without se, učit usually means to teach.

Compare:

• Učí děti. = He/She teaches children.
• Učí se plavat. = He/She is learning to swim.

And:

• plavat = to swim

So the structure is:

• učit se + infinitive
• to learn to do something

Because se is a clitic, and Czech clitics usually go in the second position in the clause.

Here the first unit is:

• naše dcera

So the clitic se comes right after that:

• naše dcera se učí plavat

Even though the dictionary form is učit se, in real sentences se often moves into second position.

This is one of the most important Czech word-order patterns to get used to.

Yes, but the meaning changes because of aspect.

• učí se plavat = she is learning to swim; the process is in progress
• naučit se plavat = to learn/master how to swim; focus on reaching the result

So:

• Naše dcera se učí plavat. = Our daughter is learning to swim.

• Naše dcera se chce naučit plavat. = Our daughter wants to learn how to swim.

• Naše dcera se naučila plavat. = Our daughter learned how to swim / managed to learn to swim.

In your sentence, učí se plavat is best because it explains the current reason for wanting to go to the seaside.