Our teacher told us, that beside modal verbs, you need to use zu + infinitive for example:

Es ist schwer, Deutsch zu lernen.

ist is not a modal verb and this is why we use zu.

But I saw these sentences recently:

Ich gehe Essen kaufen.

Isn't it suppose to be zu kaufen?

Er kommt Dan besuchen.

Isn't is suppose to be zu besuchen?

Is there any rule when you do not need zu?

  • I can't answer your question, but the sentences without zu are correct as is. However, you could use zu in these sentences: "Er kommt, Dan zu besuchen." This is an older form of um... zu and isn't used often. – Philipp Jun 10 '18 at 15:12
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    Those sentences sound to me like colloquial German, and usual rules of grammar wouldn't apply to that. Ich gehe aus, um etwas zum Essen zu kaufen, or Er kommt, um Dan zu besuchen. These of course are rather long-winded (esp the first one), so are often abbreviated in everyday usage. – Oliver Mason Jun 11 '18 at 8:27

Our teacher told us, that beside modal verbs, you need to use zu+infinitiv

That's … unfortunately not too helpful, as German makes no sharp distinction between modal verbs and others. Aside from the pretty common modal verbs müssen, können, dürfen, sollen, wollen, mögen, möchten, a number of other verbs may also be used as modals.

Gehen, kommen, sehen, hören, lassen are common "second-tier" modal verbs.

Sie geht arbeiten.

Er kommt kochen.

Sie sehen ihn fallen.

Ich hörte sie spielen.

Sie lässt ihn essen.

In general, verbs of movement, cognition, and exercise are hot candidates for modals:

Er fuhr arbeiten.

Ich fühle es kribbeln.

Sie übt radfahren.

You asked about a rule when not to use zu. The question has to be when to use zu. I've answered this question already here.

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  • Beim letzten Beispiel widerspricht mein Sprachgefühl: "Sie übt Radfahren" oder "Sie übt, Rad zu fahren". Ebenso: "Sie übt Schreiben" oder "Sie übt, zu schreiben". Ebenso mit "lernen". – mhchem Jun 11 '18 at 9:48

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