Powinieneś ćwiczyć na siłowni, jeśli na dworze jest zima.

Powinieneś means you should.

In this sentence, it is the form used when speaking to one male person informally.

Related forms:

• powinieneś — you should, to one man/boy
• powinnaś — you should, to one woman/girl
• powinniście — you should, to more than one person informally
• powinien pan / powinna pani — formal singular

So if you are talking to a woman, the sentence would start with Powinnaś...

Because Polish often drops subject pronouns when they are clear from the verb form.

In English, you must say you should.
In Polish, powinieneś already tells you:

• 2nd person
• singular
• masculine

So ty is unnecessary here. You could add ty for emphasis, but it is not normally needed.

After powinieneś, Polish uses the infinitive of the main action.

So:

• powinieneś ćwiczyć = you should exercise
• powinieneś iść = you should go
• powinieneś odpocząć = you should rest

This works similarly to English should + base verb.

So ćwiczyć is not conjugated here because powinieneś is already the finite verb-like part of the sentence.

Ćwiczyć can mean both exercise and practice, depending on context.

Examples:

• ćwiczyć na siłowni — exercise / work out at the gym
• ćwiczyć polski — practice Polish
• ćwiczyć na pianinie — practice the piano

In this sentence, because of na siłowni, it clearly means exercise or work out.

In Polish, the natural idiomatic phrase is na siłowni for at the gym.

Even though English often says in the gym, Polish usually uses na with some places and activities where English would not.

Compare:

• jestem na siłowni — I am at the gym
• ćwiczę na siłowni — I work out at the gym

With movement, you also use na, but with a different case:

• idę na siłownię — I am going to the gym

So:

• na siłowni — location
• na siłownię — destination

Siłowni is the locative singular form of siłownia.

Why locative? Because after na when it means in/at/on a place, Polish normally uses the locative.

Case pattern:

• siłownia — nominative
• na siłowni — locative

So in this sentence, na siłowni means at the gym.

Na dworze is a very common Polish expression meaning outside or outdoors.

So:

• na dworze jest zimno — it is cold outside
• dzieci bawią się na dworze — the children are playing outside

It is an idiomatic phrase, so it is best learned as a whole expression.

A more literal-sounding alternative is na zewnątrz, but na dworze is extremely common in everyday Polish.

Good question. These mean different things:

• jest zima — it is winter
• jest zimno — it is cold

So the sentence as written talks about the season:

• jeśli na dworze jest zima = if it is winter outside

That is grammatically understandable, but many learners notice it sounds a bit unusual in English and also slightly unusual in Polish.

Depending on the intended meaning, a more natural Polish sentence might be:

• Powinieneś ćwiczyć na siłowni, jeśli na dworze jest zimno.
  You should work out at the gym if it is cold outside.

Or:

• Powinieneś ćwiczyć na siłowni zimą.
  You should work out at the gym in winter.

Both jeśli and jeżeli mean if.

In most cases, they are interchangeable:

• jeśli masz czas — if you have time
• jeżeli masz czas — if you have time

The difference is mostly one of style:

• jeśli is very common and everyday
• jeżeli can sound slightly more formal or careful

In your sentence, jeśli is perfectly natural.

Because Polish normally separates a subordinate clause with a comma.

Here, jeśli na dworze jest zima is a conditional subordinate clause, so the comma is required:

• Powinieneś ćwiczyć na siłowni, jeśli na dworze jest zima.

If you reverse the order, you still need a comma:

• Jeśli na dworze jest zima, powinieneś ćwiczyć na siłowni.

Yes. Polish word order is more flexible than English.

A very natural alternative is:

• Jeśli na dworze jest zima, powinieneś ćwiczyć na siłowni.

This puts the condition first, which often feels very natural.

The original version:

• Powinieneś ćwiczyć na siłowni, jeśli na dworze jest zima.

Both are correct. The difference is mostly about emphasis and information flow, not basic meaning.

A rough breakdown is:

• Powinieneś — you should
• ćwiczyć — exercise / work out
• na siłowni — at the gym
• jeśli — if
• na dworze — outside
• jest zima — it is winter

One thing English speakers often notice is that Polish does not need a dummy subject like it in weather/season expressions the way English does.

So English says:

• it is winter
• it is cold

But Polish can simply say:

• jest zima
• jest zimno

It is grammatically correct, but depending on the intended meaning, many speakers might choose a slightly different phrasing.

More natural options could be:

• Powinieneś ćwiczyć na siłowni, jeśli na dworze jest zimno.
  If it is cold outside, you should work out at the gym.

• Powinieneś ćwiczyć na siłowni zimą.
  You should work out at the gym in winter.

• Kiedy jest zima, powinieneś ćwiczyć na siłowni.
  When it is winter, you should work out at the gym.

So the original sentence is useful for learning grammar, but in real life a native speaker might pick a smoother version depending on what exactly they want to say.