Гравець кидає кубик і бере фішку.

Гравець is the subject of the sentence, so it is in the nominative case.

• гравець = player
• Nominative is the basic dictionary form for a noun when it is doing the action.

Here, the player is the one who throws and takes, so гравець stays in the nominative.

Because it is the direct object of бере (takes), so it goes into the accusative case.

For many feminine nouns ending in -а, the accusative singular changes like this:

• фішка → фішку
• книга → книгу
• карта → карту

So бере фішку literally means takes a token/piece, with фішку showing that it is the thing being taken.

Because кубик is a masculine inanimate noun, and in Ukrainian, the accusative singular of masculine inanimate nouns is usually the same as the nominative singular.

So:

• nominative: кубик
• accusative: кубик

That is why you get:

• кидає кубик

but:

• бере фішку

The two objects are both in the accusative, but they look different because they belong to different noun types.

Кидає means he/she throws or is throwing. The ending -є marks:

• 3rd person singular
• present tense

So:

• я кидаю = I throw
• ти кидаєш = you throw
• він / вона кидає = he / she throws

Since гравець is a singular noun, the verb must also be singular: кидає.

Бере is the present-tense 3rd person singular form of брати (to take).

This verb changes in a way you just have to learn:

• брати = to take
• я беру = I take
• ти береш = you take
• він / вона бере = he / she takes

So бере is the normal present-tense form here, even though it does not look exactly like the infinitive.

Yes. In Ukrainian, the present tense can describe:

• something happening right now
• a habitual action
• a step in a rule, instruction, or narrative

So Гравець кидає кубик і бере фішку can mean:

• The player throws the die and takes a token right now, or
• A player throws the die and takes a token as part of the game procedure.

Context tells you which reading is intended.

The verbs here are кидати and брати, which are imperfective.

Imperfective verbs are commonly used for:

• repeated actions
• general statements
• describing what happens in a process or rule
• present tense

That fits a sentence like this very well, especially in a game context.

The rough perfective partners would be:

• кидати → кинути
• брати → взяти

But perfective verbs normally do not have a true present-tense meaning in Ukrainian. Their present-looking forms usually refer to the future.

So for a rule or ongoing description, кидає and бере are the natural choice.

Ukrainian word order is more flexible than English because case endings show grammatical roles.

The neutral, most straightforward order here is:

• Гравець кидає кубик і бере фішку.

But other orders are possible for emphasis, for example:

• Кубик кидає гравець і бере фішку.
  This is grammatically possible, but it sounds marked and unusual without a special context.

In normal usage, especially for learners, it is best to keep the basic order:

• subject + verb + object

Because Ukrainian has no articles.

English needs words like:

• a
• an
• the

Ukrainian does not. So гравець, кубик, and фішку can mean:

• a player / the player
• a die / the die
• a token / the token

Context tells you which one is meant.

Yes. In a board-game context, кубик commonly means a die.

This can be confusing because English often uses dice very loosely, even for one item. But traditionally:

• die = one
• dice = more than one

In Ukrainian:

• кубик = one die
• кубики = dice

So кидає кубик literally means throws a die.

A few useful pronunciation notes:

• г in Ukrainian is usually like a voiced English h, not a hard English g
• є often sounds like ye or softens the preceding consonant
• и is not the same as English ee

Approximate pronunciation:

• Гравець ≈ hra-VEHTS
• кидає ≈ ky-DA-ye
• кубик ≈ KOO-byk
• бере ≈ be-REH
• фішку ≈ FEESH-koo

Useful stress:

• граве́ць
• кида́є
• ку́бик
• бере́
• фі́шку

These are only approximations, but they are a good starting point.

Often, yes. Both і and та can mean and.

So you may see:

• Гравець кидає кубик і бере фішку.
• Гравець кидає кубик та бере фішку.

In many contexts, і is the most neutral and common choice. Learners should definitely know і first.

Also, in real speech and writing, Ukrainian sometimes chooses і / й / та partly for sound and style. But in this sentence, і is completely natural.