I consider my German intermediate and began to read Harry Potter to improve my knowledge, and I stumbled upon this sentence:
... als Harry ihm und Hermine zu erzählen begann, ...
Shouldn't it be begann zu erzählen? Or perhaps if you only have one verb after the zu and nothing else you can switch it? Or is it allowed only if the first word is beginnen? Could I say "Er zu schreien versuchte"?
You have "normal" word order in main sentences and a special word order in subclauses. The verb can have three positions
1 Er kam zu spät. -Verb in position 2 after subject.
2 ... weil er zu spät kam. - Subclause, verb in end position.
3 Deswegen kam er zu spät.- The position before the subject "er" is occupied with deswegen. Verb in front position before subject.
The word order is one of the difficulties of German. Children suck it up with the mother milk. For learners it is difficult and you need a lot of training to master word order as in 2 and 3. I would recommend to use a grammar and to collect a lot of example material. Finally this change of the verb position must be done automatically.
This is not only word (verb) order. In fact, als Harry begann, ihm und Hermine zu erzählen, … (was sich vorgetragen hatte) is also a perfectly valid sentence, contradicting the subordinate clause verb order argument.
This is because ihm und Hermine zu erzählen is an extended infinitive. Those can get miles longer than the actual sentence itself – read some Kafka if you want to know the extremes. Because these can get extremely long, it is permissable to pull the verb in front of the extended infinitive, so that readers won't have forgotten the original subordinate clause by the time they reach the verb.
The real reason why the translators chose als Harry ihm und Hermine zu erzählen begann is style. If the extended infinitive is placed after the finite verb, it requires a pair of commas separating it from the sentence, causing readers to make pauses and 'ripping' the sentence 'apart'. On the other hand, the whole thing is short enough that the begann can be put at the end without loosing context. This also allows Harry, Hermione and 'him' (I'm assuming Ron?) to be perceived much more as a group and the story that's being told later as more separate … and German teachers could go into interpretational depths of that to infinity. And of course, since the sentence continues past the part you quoted, it's nicer to keep it short.
Compare:
Während ich diese Antwort schrieb, überlegte ich mir fieberhaft, ob ich nicht ein weiteres, ausgiebigeres Beispiel auszuformulieren und zu gestalten in der Lage gewesen wäre.
With:
Während ich diese Antwort schrieb, überlegte ich mir fieberhaft, ob ich nicht in der Lage gewesen wäre, ein weiteres, ausgiebigeres Beispiel auszuformulieren und zu gestalten.
You see that pulling forward the extended infinitive even works with longer ones (in this case there are two verbs with a zu, connected by an und). But in this case, most writers would use the latter, because it reads itself so much easier.
Als Harry ihm und Hermine begann zu erzählen is wrong due to the word order issue rogermue pointed out. To analyse it in its entirety, consider that ihm und Hermine does not belong to begann (beginnen does not take a dative object) but rather to erzählen. However, erzählen is an infinitive with zu here, so ihm und Hermine has to be close to, nigh unseparable with the zu erzählen part. This cannot be torn apart by anything belonging to the outer clause. It could only be torn apart by adverbs further specifying how the telling was done. Thus, begann can only be before or after the entire complex, never in between.
