Short answer: To be sure both, and check with the customs of the place you're flying to if it's an international flight.
The airport security will have it's own rules, probably based on national or international standards. The airline will also follow those rules but may impose additional rules or requirements. Finally the country you are flying to may have rules regarding import and declaration of different things.
The airline will likely know the local airport rules, but they may not have completely up to date information. The airport staff will not necessarily know about specific airline rules. And neither of them will care if you're going to get your Vegemite confiscated when they land, because all they care about is the safety of the airport, aircraft and getting you to the destination (and through immigration there).
Fortunately you don't specifically need to contact anyone as all this information is generally on the internet. If you're carrying something unusual then you need to check, and it can't hurt to check over the phone just in case the rules have changed.
For examples, here's Cathay Pacific's Prohibited Items list you'll want to cross-check the Liquids / Gels / Aerosols restrictions page too. And here's Perth Airport page covering both. Finally here's the generic Australia page on the same topic.
For your list, Toothpaste, Mouthwash and (probably) Vegemite are all going to count as 'Liquids, aerosols or gels' therefore (from my last link):
- Liquid, aerosol or gel products (such as toiletries) must be in containers of 100 millilitres or less (broadly equivalent to 100 grams or less).
- The containers must be carried in one transparent, resealable plastic bag, like the one in the image below.
- The four sides of the bag's sealed area must add up to no more than 80 centimetres (e.g. 20x20 cm or 15x25 cm).
There's more on the page, but that's the basics. One bag, resealable, matching those sizes, each individual thing in containers of 100ml of less. That's for cabin baggage, all of those should be fine in checked baggage if you have bigger sizes.
Chocolate should be fine in carry-on or checked bagged as long as it's a reasonable amount, I guess there might be a country somewhere that controls the import of chocolate but I can't find one.
I'm guessing your flight is domestic, but for completeness here's the Australian Customs page on Food, Plants and Animals which may apply.