There is no requirement in the UK primary school English curriculum for pupils to be able to recognise different types of adverb. They may be required to identify an adverb in situe and will certainly be expected to use adverbs in their own writing. You can see details of the most commonly encountered adverbs here:
The following are types of adverbs which are not quite as obvious but, nevertheless, worth knowing about..
Evaluative adverbs (surprisingly) and viewpoint adverbs (personally)
We put some adverbs outside the clause. They modify the whole sentence or utterance. Evaluative and viewpoint adverbs are good examples of this…
The supermarkets own brand make-up, surprisingly, gave results as good as the major brands.
Personally, I think HP is the best of all the brown sauces on offer.
Linking adverbs (then, however) demonstrate a relationship between two clauses or sentences (e.g. a sequence in time, cause and effect, contrast between two things):
I left my house in the morning then I went to pick up Leanne at her house.
We talked until the early hours and consequently I overslept the next morning.
It is raining on the East coast. However, today the sun is shining in Blackpool.