На рынке продавщица посоветовала взять сладкую дыню и спелый арбуз.

Because на is the normal preposition for being at a market, fair, station, concert, and similar public places/events.

• на рынке = at the market
• на рынке uses the prepositional case because it describes location

Compare:

• на рынок = to the market, with motion
• на рынке = at the market, with location

So here рынке is the prepositional form of рынок.

Продавщица means female seller / saleswoman / shop assistant. It tells you the seller is a woman.

A related word is продавец:

• продавец = seller, often male or gender-neutral in some contexts
• продавщица = specifically female seller

In this sentence, the word choice matters because it also affects the verb form: the verb is feminine singular because продавщица is feminine.

This is the past tense feminine singular form of посоветовать.

In Russian past tense, the verb agrees with the subject in gender and number:

• посоветовал = he advised
• посоветовала = she advised
• посоветовало = it advised
• посоветовали = they advised

Since the subject is продавщица, a feminine singular noun, the verb must be посоветовала.

Because Russian commonly uses the pattern:

посоветовать + infinitive

This means to advise/recommend doing something.

So:

• посоветовала взять = advised to take / recommended taking

You can also include the person being advised:

• продавщица посоветовала мне взять... = the seller advised me to take...

In your sentence, that indirect object is simply omitted because it is understood from context.

This is an aspect question.

• взять = perfective
• брать = imperfective

Here взять is used because the seller is recommending a specific, completed action: choosing or buying these particular fruits.

So посоветовала взять sounds like:

• take this melon and this watermelon
• get these items

If you used брать, it would sound more like an ongoing process, a repeated action, or a more general habit, which is not the main idea here.

Because дыню is the direct object of взять, so it must be in the accusative case.

The basic form is:

• сладкая дыня = sweet melon

But after взять:

• взять сладкую дыню = to take a sweet melon

Both the adjective and the noun change to match the accusative feminine singular:

• сладкая → сладкую
• дыня → дыню

Because арбуз is masculine singular inanimate, and for masculine inanimate nouns, the accusative singular is the same as the nominative singular.

So:

• спелый арбуз = ripe watermelon
• взять спелый арбуз = take a ripe watermelon

The form does not change visibly here.

This is different from masculine animate nouns, where the accusative usually matches the genitive:

• хороший продавец = good seller
• вижу хорошего продавца = I see a good seller

Because they belong to different noun types.

• дыня is feminine, so its accusative singular changes clearly: дыню
• арбуз is masculine inanimate, so its accusative singular looks like the nominative: арбуз

So both objects are in the accusative, but Russian shows that case differently depending on gender and animacy.

Literally, взять means to take. But in shop or market contexts, it often means something closer to:

• get
• take
• pick
• buy

So in this sentence, the natural English meaning is often buy or get, even though the Russian verb is взять.

This is very common in everyday Russian:

• Я возьму этот сыр. = I’ll take/get this cheese.
• In context, that often means I’ll buy this cheese.

Russian often leaves out the indirect object when it is obvious from context.

So:

• продавщица посоветовала взять... = the seller advised/recommended taking...
• продавщица посоветовала мне взять... = the seller advised me to take...

Both are possible. In your sentence, the person being advised is simply understood.

Yes. Russian word order is much more flexible than English word order.

The sentence starts with На рынке, which sets the scene first:

• На рынке продавщица посоветовала взять...

That sounds natural and emphasizes the setting: At the market...

You could also say:

• Продавщица на рынке посоветовала взять...

This is also grammatical, but the focus shifts slightly toward the seller rather than the location.

Russian word order often changes emphasis rather than basic meaning.

Russian has no articles like a and the.

So:

• продавщица can mean a seller or the seller
• дыню can mean a melon or the melon
• арбуз can mean a watermelon or the watermelon

Context tells you which meaning is intended. In a sentence like this, English usually chooses the seller if the person is already identifiable in the situation, but Russian does not mark that with an article.