Learn Hardcore Norwegian: Jeg fyller handlekurven med grønnsaker, men passer på å ikke gå over budsjettet. - I fill the shopping basket with vegetables, but make sure not to go over the budget.

Handlekurv means shopping basket in its basic (indefinite) form: en handlekurv.
Norwegian adds -en to make the definite singular form: handlekurven = the shopping basket.

In this sentence you’re talking about a specific basket you’re currently using, so the definite form handlekurven (the basket) is natural.

Structure:

en handlekurv – a shopping basket
handlekurven – the shopping basket

Why can the second clause start with men passer på and not repeat jeg?

In Norwegian, when you have the same subject in two clauses joined by og, men, eller, etc., you can drop the repeated subject in the second clause.

So instead of:

Jeg fyller handlekurven med grønnsaker, men jeg passer på å ikke gå over budsjettet.

you can (and usually do) say:

Jeg fyller handlekurven med grønnsaker, men passer på å ikke gå over budsjettet.

It’s understood that jeg is still the subject in the second part.

What does passer på å mean here? Is it the same as prøver å?

Passe på å + infinitive means something like make sure to / take care to / be careful to.
In this context:

passer på å ikke gå over budsjettet ≈ make sure not to go over the budget / be careful not to go over the budget.

Prøver å simply means try to:

prøver å ikke gå over budsjettet = try not to go over the budget.

Both are possible, but:

passer på å focuses on being careful/attentive.
prøver å focuses on the effort of trying, not necessarily on attentiveness.

Why is it passe på and not just passe?

Passe på is a phrasal (particle) verb in Norwegian; the på is part of the expression and you normally need it for this meaning:

passe på å gjøre noe – make sure to do something / be careful to do something.

Passe alone has other meanings:

Det passer – That suits/works/is convenient.
Han passer barna – He looks after the children.

So for the sense “make sure / be careful to”, you say passe på (å), not just passe (å).

Why is ikke placed between å and gå (…å ikke gå over)? Could it be å gå ikke over or ikke å gå over?

The normal word order is:

å ikke + infinitive → å ikke gå over

You cannot say å gå ikke over – the negation ikke does not go between the verb and its particle/preposition like that in infinitive phrases.

Ikke å gå over is possible in some special emphatic or contrastive contexts (e.g. Det viktigste er ikke å gå over budsjettet – The most important thing is not to go over the budget). But after passe på å, the natural, neutral order is:

passe på å ikke gå over budsjettet.

Why is it med grønnsaker and not av grønnsaker?

With the verb fylle (opp) in the sense of filling a container with something, Norwegian normally uses med:

fylle glasset med vann – fill the glass with water
fylle handlekurven med grønnsaker – fill the basket with vegetables

Av is used more for made of/from:

en stol av tre – a chair made of wood
en saus av tomater – a sauce made from tomatoes

So for putting things into a container, use med, not av.

Why is grønnsaker plural and indefinite (not grønnsakene)?

Grønnsaker is the plural indefinite form: vegetables in general.

grønnsaker – vegetables (in general, not specified which ones)
grønnsakene – the vegetables (specific ones already known from context)

Here the idea is simply that you’re filling the basket with vegetables in general, not some particular set that both speaker and listener are already focusing on. So grønnsaker (indefinite plural) is the natural form.

What’s the difference between fylle handlekurven and fylle opp handlekurven?

Both are correct, but there’s a nuance:

fylle handlekurven – fill the basket (neutral; just putting things into it).
fylle opp handlekurven – fill the basket up, often implying more completely / to the top.

In everyday speech, fylle opp can suggest that the basket becomes rather full, while fylle is more neutral about how full it ends up.

Why is budsjettet definite? Could we say just gå over budsjett?

Budsjettet is definite singular: the budget (a specific one – usually my budget in this context).

budsjett (indefinite) – a budget, budget in general
budsjettet (definite) – the budget (the one we have set)

In this kind of personal sentence, Norwegians almost always use the definite form, because there is a specific, known budget involved.

You can say gå over budsjett in some set expressions (often in business contexts: go over budget), but in everyday personal shopping talk, gå over budsjettet sounds more natural.

What does gå over budsjettet literally mean, and is there a more formal alternative?

Literally:

gå over budsjettet = go over the budget, i.e. spend more than the budget allows.

A more formal or written alternative is:

ikke overskride budsjettet – not exceed the budget.

In normal spoken Norwegian, gå over budsjettet is more common and more colloquial; overskride budsjettet sounds more formal and technical.

Why is the verb tense fyller / passer på (present), and how does that compare to English “I am filling / I’m making sure”?

Norwegian uses the simple present (fyller, passer på) to cover both:

English simple present (I fill / I make sure)
and
English present continuous (I am filling / I’m making sure).

So:

Jeg fyller handlekurven… can mean I am filling the basket… (right now) or I fill the basket… (habitually), depending on context.
Norwegian normally does not need an extra auxiliary like am/is/are to show continuous actions in the present.