Dvouhra i čtyřhra začínají krátkým podáním nebo dlouhým podáním.

Because the subject is actually two things together: dvouhra i čtyřhra = singles and doubles.

Each noun by itself is singular, but together they form a compound subject, so Czech uses the plural verb:

• dvouhra začíná = singles begins
• čtyřhra začíná = doubles begins
• dvouhra i čtyřhra začínají = singles and doubles begin

So začínají is the correct 3rd person plural form.

Here i means something like and also / both ... and ....

In this sentence, dvouhra i čtyřhra means:

• both singles and doubles
• or simply singles and doubles

Czech i can sometimes mean also, even, or function a bit like and depending on context. Here it links the two nouns and emphasizes that the statement applies to both.

The infinitive is začínat.

• začínat = to begin / to start
• začínají = they begin / they start

In this sentence, it works like English start with:

• začínají krátkým podáním = they begin with a short serve

So the Czech structure is very close in meaning to English, even though the grammar uses a case ending instead of the separate word with.

Because after začínat in the sense of begin with, Czech often uses the instrumental case.

So:

• krátké podání = short serve
• krátkým podáním = with a short serve

and

• dlouhé podání = long serve
• dlouhým podáním = with a long serve

Both the adjective and the noun change form to match the instrumental singular neuter:

• krátké → krátkým
• dlouhé → dlouhým
• podání → podáním

Podáním is instrumental singular.

It is used because Czech often expresses with what something begins by using the instrumental case after začínat.

So:

• začínat něčím = to begin with something

Examples:

• začít otázkou = to begin with a question
• začínat větou = to begin with a sentence
• začínat podáním = to begin with a serve

So in your sentence, the instrumental shows the means or element something starts with.

Yes, Czech could leave it out.

A shorter version would be:

Dvouhra i čtyřhra začínají krátkým nebo dlouhým podáním.

That is perfectly natural and usually more economical.

The full version with repetition:

krátkým podáním nebo dlouhým podáním

is still grammatical, but it is a bit more explicit or formal. Repeating the noun can make the sentence feel clearer, especially in instructional or rule-based language.

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

So Czech simply says:

• krátkým podáním
• literally something like short serve / with short serve

Whether English should translate it as a short serve, the short serve, or just short serve depends on context.

This is normal in Czech. Learners often have to get used to understanding definiteness from the situation rather than from an article.

• dvouhra is feminine
• čtyřhra is feminine
• podání is neuter

You can often recognize this from the endings:

• nouns ending in -a are often feminine
• nouns ending in -í are often neuter

That is why, if the subjects were singular, you would get:

• dvouhra začíná
• čtyřhra začíná

But since the two feminine nouns are joined together, the verb becomes plural:

• dvouhra i čtyřhra začínají

Not completely. Czech word order is fairly flexible.

The given order:

Dvouhra i čtyřhra začínají krátkým podáním nebo dlouhým podáním.

is neutral and natural. It starts with the topic, dvouhra i čtyřhra, and then says what happens.

Other word orders are possible, for example to change emphasis:

• Krátkým podáním nebo dlouhým podáním začínají dvouhra i čtyřhra.

This version puts more emphasis on how they begin. It is grammatical, but less neutral in everyday use.

So the sentence order you were given is probably the most straightforward one.

Yes. Nebo is the standard Czech word for or.

So:

• krátkým podáním nebo dlouhým podáním = with a short serve or a long serve

In many contexts, nebo is just a plain, ordinary or. Sometimes context decides whether it is exclusive or inclusive, just like in English.

These words can look intimidating at first.

• dvouhra is roughly dvoh-ra, with dvou sounding like dvoh
• čtyřhra is harder: roughly chti-rh-ra or chtyr-hra, depending on how carefully it is pronounced

A few tips:

• č sounds like ch in church
• ř is the famous Czech sound with no exact English equivalent
• in normal speech, clusters can sound smoother than they look on paper

If čtyřhra feels difficult, that is normal even for learners at a fairly good level. It is a very Czech-looking word.