На старой лужайке рядом с муравейником мы увидели ромашку и маленький бутон лилии.

Because after на when it means on / at a location, Russian usually uses the prepositional case.

• лужайка = lawn / grassy clearing
• Prepositional singular: лужайке
• The adjective must match it:
  • старая → старой

So:

• на старой лужайке = on the old lawn

If на meant motion onto something, Russian would use a different case, but here it describes location, so prepositional is correct.

Because старой agrees with лужайке.

лужайка is:

• feminine
• singular
• prepositional case

So the adjective must also be:

• feminine
• singular
• prepositional

That gives:

• старая → nominative
• старой → prepositional

Russian adjectives always agree with the noun in gender, number, and case.

Because рядом с means next to / beside / near, and it requires the noun after it to be in the instrumental case.

So:

• муравейник = anthill
• instrumental singular: муравейником

That is why you get:

• рядом с муравейником = next to the anthill

This is just something to memorize:
рядом с + instrumental

It is instrumental singular.

The base noun is:

• муравейник

For many masculine nouns, the instrumental singular ends in -ом or -ем:

• муравейник → муравейником

So the ending changes because of the preposition phrase рядом с.

Увидели is:

• past tense
• plural
• from the verb увидеть = to see / to catch sight of
• perfective aspect

It is plural because the subject is мы = we.

So:

• мы увидели = we saw

In the past tense, Russian verbs agree with the subject in number and, in the singular, also in gender.

Because увидели is perfective, and it usually means a completed event: we saw / we noticed / we caught sight of.

Compare:

• мы видели = we were seeing / we saw in a more general or ongoing sense
• мы увидели = we saw as a single completed event

In this sentence, the idea is that we noticed something specific at that moment, so увидели fits very naturally.

Yes, very often.

Russian often drops subject pronouns when the meaning is clear from the verb form. So:

• Мы увидели ромашку...
• Увидели ромашку...

Both can work, depending on context.

Including мы can make the subject clearer or slightly more explicit, but it is not always necessary.

Because ромашку is the accusative singular form, and here it is a direct object of the verb увидели.

• ромашка = nominative singular
• ромашку = accusative singular

For many feminine nouns ending in -а, the accusative singular changes to -у:

• ромашка → ромашку

So:

• мы увидели ромашку = we saw a camomile / daisy

Because бутон is also a direct object, and it is masculine singular inanimate.

For masculine inanimate nouns, the accusative singular is usually the same as the nominative singular.

So:

• nominative: маленький бутон
• accusative: маленький бутон

That is why nothing changes here.

If the noun were masculine animate, the form would look different.

Because лилии is in the genitive case after another noun.

Russian often uses this pattern to mean of something:

• бутон лилии = the bud of a lily

So:

• бутон = bud
• лилия = lily
• genitive singular of лилия = лилии

This is a very common structure in Russian:

• книга учителя = the teacher’s book / the book of the teacher
• бутон лилии = a lily bud / the bud of a lily

There are two adjective + noun pairs:

1. старой лужайке

   • noun: feminine, singular, prepositional
   • adjective matches: старой

2. маленький бутон

   • noun: masculine, singular, accusative
   • but because it is inanimate, accusative looks like nominative
   • adjective matches: маленький

So the adjective endings are different because the nouns have different:

• gender
• case
• grammatical role

Because Russian word order is flexible, and putting the location first helps set the scene.

The sentence starts with the setting:

• On the old lawn next to the anthill...

Then comes the main action:

• we saw...

This is very natural in Russian. English can do something similar too, but Russian allows this kind of fronting even more freely.

Russian has no articles like a/an/the.

So forms like:

• лужайке
• муравейником
• ромашку
• бутон лилии

do not themselves tell you whether something is a or the.

You figure that out from:

• context
• word order
• whether the thing is already known in the conversation
• what sounds natural in English

So depending on context, ромашку could be:

• a daisy
• the daisy

In your translation, the intended meaning has already been given, but the Russian sentence itself does not mark articles directly.

It usually means next to, beside, or nearby, often with a fairly close sense.

So:

• рядом с муравейником = next to / beside the anthill

It often suggests more closeness than some other words for near, though in real usage there can be overlap.

For example:

• рядом с домом = next to the house / by the house
• около дома = near the house

Both can be translated as near, but рядом с often feels a bit more physically adjacent.

Yes. A learner may want to note the stressed syllables:

• На старо́й лужа́йке
• ря́дом с мураве́йником
• мы уви́дели
• рома́шку
• и ма́ленький буто́н ли́лии

A few helpful points:

• ё is not written here because there is no ё in these words, but in Russian stress is still important.
• увидели is stressed on ви́.
• муравейником has stress on ве́й.
• лилии can feel tricky because of the repeated и sounds.

Reading the sentence slowly in chunks helps:

• На старо́й лужа́йке
• ря́дом с мураве́йником
• мы уви́дели
• рома́шку и ма́ленький буто́н ли́лии