Utáhnu šroub na řídítkách.

Because Czech usually drops subject pronouns when the verb ending already tells you the person and number.

• Utáhnu already means I will tighten
• so adding já is usually unnecessary

You can add já for emphasis or contrast:

• Já utáhnu šroub, ty zkontroluješ brzdy.
  = I’ll tighten the screw, you check the brakes.

The infinitive is utáhnout, which is a perfective verb. In Czech, present-tense forms of perfective verbs usually refer to the future.

So:

• utáhnu = I will tighten
• not normally I am tightening

If you want an ongoing or habitual present meaning, Czech uses an imperfective verb instead, such as utahovat:

• Utahuju šroub. = I’m tightening the screw.
• Budu utahovat šroub. = I will be tightening the screw.

The infinitive is utáhnout.

Its conjugation changes the stem a little:

• utáhnout
• utáhnu
• utáhneš
• utáhne
• etc.

So yes, this is a normal verb pattern, even though it does not look as simple as just removing -out and adding an ending.

Šroub is the direct object, so it is in the accusative case.

But for many masculine inanimate nouns, the accusative singular looks the same as the nominative singular. So:

• nominative: šroub
• accusative: šroub

That is why the form does not visibly change.

Compare that with a masculine animate noun, where the accusative often looks different:

• pes → vidím psa

Because na can take different cases depending on the meaning:

• na + locative = location, on
• na + accusative = movement onto something

Here the meaning is location: the screw is on the handlebars, so Czech uses the locative:

• na řídítkách

Compare:

• Mám ruce na řídítkách. = My hands are on the handlebars.
• Dám ruce na řídítka. = I’ll put my hands onto the handlebars.

In Czech, řídítka is normally a plural-only noun. That means it refers to one object, but grammatically it behaves like a plural word.

This is similar to some English words like scissors or pants.

So you get forms like:

• dictionary form: řídítka
• genitive: řídítek
• locative: na řídítkách

Even though it is one handlebar assembly, Czech treats it as plural.

Most naturally, it describes the screw:

• the screw on the handlebars

So the sentence is understood as:

• I’ll tighten the screw that is on the handlebars

If you wanted to make that relationship completely explicit, you could say:

• Utáhnu šroub, který je na řídítkách.

But in normal Czech, the shorter version is very natural.

Yes. Czech word order is fairly flexible, and changes often affect emphasis rather than basic meaning.

The sentence

• Utáhnu šroub na řídítkách.

is a natural neutral order.

You could also say:

• Na řídítkách utáhnu šroub.

That puts more focus on the handlebars as the place being discussed.

So the basic meaning stays the same, but the information structure changes.

Because Czech has no articles.

So šroub can mean:

• a screw
• the screw

and the exact meaning comes from context.

If Czech speakers want to be more explicit, they can use other words such as demonstratives:

• ten šroub = that / the screw

But in many sentences, no extra word is needed.

A few useful points:

• ř is the famous Czech sound; it is like an r with added friction
• í and á are long vowels
• ch is one sound, like the ch in German Bach or Scottish loch
• Czech stress is usually on the first syllable of the word

So roughly:

• řídítkách → stress on ří

If ř is difficult, that is completely normal for learners. Even getting close with a strong r-like sound at first is a good start.