Learn Italian: Questo cane è molto vecchio.

Questo means "this" for masculine singular nouns, and questa is for feminine ones. Because cane (dog) is a masculine noun, we must use questo.

Cane ends in -e, so how can I tell if it is masculine or feminine?

Nouns ending in -e can be either masculine or feminine, and you usually just have to memorize which one they are. However, in this sentence, the words around it give you a huge clue! Both the word for "this" (questo) and the adjective "old" (vecchio) end in -o, which confirms that cane is masculine.

Will molto ever change to molti or molta in sentences like this?

Not when it means "very". When molto is used to describe an adjective (like being "very" old), it never changes its ending. It only changes endings when it describes a noun to mean "a lot of" or "many" (like molti cani for "many dogs").

Why does vecchio end in -o?

In Italian, adjectives have to match the gender and amount of the noun they describe. Since cane is a single masculine noun, the adjective vecchio (old) must take the masculine singular ending -o.