Learn Polish: Kupuję szynkę i ser. - I am buying ham and cheese.

They are both in the accusative case because they are the direct objects of the verb (the things being bought). Szynka is a feminine noun, and feminine nouns change their ending from -a to -ę in the accusative. Ser is a masculine inanimate noun, and these don't change their endings in the accusative case; they look exactly like the dictionary form.

How do I know kupuję means "I am buying" and not "he is buying"?

In Polish, the verb ending tells you who is doing the action, so the pronoun ja (I) is usually dropped. The -ę (with the tail) on kupuję specifically means "I". If it were "he is buying", it would be kupuje with a standard -e at the end.

The dictionary says "to buy" is kupić, so where does kupuję come from?

Kupić is a perfective verb, used for completed actions. To say "I am buying" right now, you must use the imperfective version of the verb, which is kupować. Verbs that end in -ować drop that ending and add -uję for the "I" form in the present tense.

Could I use a instead of i here to mean "and"?