there is / are

There is/are ...

"there" (or "here") in this kind of sentence is actually not the real subject of the sentence. It is called expletive constructions (填補). The real subject is after the verb (to be).

The rule of subject-verb agreement is:
With these constructions, the subject follows the verb but still determines the number of the verb.

That is to say, if the subject is plural, we use a plural verb (are, were).
Both of your sentences have plural subjects, so we use plural verbs.

(1) There are a bottle and a book on the desk.
==> A bottle and a book are on the desk.

(2) There are an orange and two apples in the basket.
==> An orange and two apples are in the basket.

用is還是are是按所要表示東西是單數還是眾數來決定

There (is/are) a bottle and a book on the desk.
這句的東西是a bottle單數...a book單數
there後是單數便用is了
There is a bottle and a book on the desk.


There (is/are) an orange and two apples in the basket.
這句的東西是an orange單數...two apples 眾數
there後是單數便用is了
There is an orange and two apples in the basket.

再加多幾句

There are oranges and apples in the basket.

There are oranges and a bannar in the basket.

英文文法有些是很直接(細微)的結構關係，文法不是講求宏觀的統計。