Learn German: Wir wollen eine neue Band sehen.

Because it comes directly before the noun it describes. The noun Band is feminine, and it is the direct object (Accusative case) of the sentence. After the feminine indefinite article eine, the adjective takes the ending -e, making it neue.

Why is sehen at the very end of the sentence?

This is the standard word order when using a modal verb like wollen (to want). The conjugated modal verb (wollen) takes the second position in the sentence, and the main action (sehen) gets pushed to the very end in its basic infinitive form.

Is Band a German word, and how do we know its gender?

Band (meaning a musical group) is a loanword borrowed directly from English. Like many imported words, its gender just has to be memorized: it is feminine (die Band). Watch out, because German also has native words spelled the same way but with different genders and meanings, like das Band (the ribbon) and der Band (the volume of a book).

Why isn't there a zu before sehen? In English we say "want to see".

In German, modal verbs like wollen (to want) connect directly to the main verb without needing zu (to). You simply leave the main verb in its dictionary form and place it at the end of the clause.