Do you have to?
Wanting to use double-nested lists is usually a sign that something has gone in your rhetoric and you should think about how to phrase to avoid this. You could use bulleted lists or make three sentences.
I need to get some ingredients for the party. For the fruit salad I'll need bananas, apples and oranges. Then I'll get some crisps and crackers for snacking, and for drinks I'll get some bottled water and cola.
(Two language/cultural notes - in Britain, chips are thick pieces of fried potato, served hot as part of a meal, eg with fried fish. "Soda" is not often sold, and then only as a mixer. You might mean "bottled water", which may be sparkling or still. Soda doesn't mean a generic bubbly drink, like it does in parts of the USA, without context "soda" can mean "washing soda" or even "baking soda")
Oxford commas are optional, which means it is correct to include them or avoid them.
Now if you must use the nested list use a full colon to introduce the list. Oxford commas are optional before "and" (I've omitted them here) but don't use commas for a two-item list. Using a semicolon after "snacking" helps to clarify that a new high-level item is starting.
I need to buy ingredients for the party: bananas, apples and oranges for fruit salad; chips and crackers for snacking; and soda, water and cola for drinks.
This is one reasonable punctuation. But there are other ways that are just as correct.