It says "Lucus quidem ille et haec Arpinatium quercus agnoscitur", but shouldn't "agnoscuntur" be used instead? As it says "the grove and those oak trees of Arpinums are recognized(by me)". I'm still struggling with the basics, so maybe I'm simply missing something important here.
You are correct. If you as a beginner wrote agnoscitur, it would be wrong.
There are two possible explanations.
Grammatical Sometimes the verb agrees with the nearest subject, and in a perfect passive this might include gender as well as number. Quercus is feminine; so haec quercus is Nominative Singular; so, 'haec quercus agnota est, would be the Perfect.
Figura There is a figure of speech known as 'hendiadys,' in which the subject is described in two ways. 'The grove, and' then more precisely 'the oak tree, is known.'
Wikipedia on Hendiadys gives a Latin example: "There are many examples in Virgil's Aeneid, e.g., Book 1, line 54: vinclis et carcere, literally translated as "with chains and prison" but the phrase means "with prison chains"."
How do you know that "quercus" is plural? If it's singular and the grove belongs to someone else (ille lucus = that grove yonder i.e. some distance away, possibly belonging to another) then the one, single oak tree, of the Arpinums, is recognised (= "agnoscitur"). The masculine, singular, nominative demonstrative pronoun "ille" = "that yonder"; as opposed to masc., sing., demonstr., nom., "is" = "that (near to me)". Similarly: feminine, singular, nominative, demonstrative pronoun, "haec" = this", indicating one oak tree; plural form, "hae", would have indicated "trees", plural.