Code or address?

Which airport and where are you going?

Do you mean “post code”? Where are you going, where are you starting from?

This is the Cardiff bus interactive route planning map-
https://www.cardiffbus.com/plan-a-journey?origin%5Bname%5D=Cardiff+Airport+Access+Road&origin%5Blocation%5D%5Blat%5D=51.399484&origin%5Blocation%5D%5Blon%5D=-3.337306&destination%5Bname%5D=Cardiff+Queen+Street+Station&destination%5Blocation%5D%5Blat%5D=51.482002&destination%5Blocation%5D%5Blon%5D=-3.170293&time%5Bwhen%5D=now

Did you try Traveline or Traveline Cymru (the welsh specific version)

Using Traveline.cymru

By looking at Google Maps with the postcode you gave - CF24 3BA - I can see that the place is a short walk from Cardiff Queen Street station.

Then looking at the airport website I see that they have a £1-00 shuttle bus to the train from the airport which takes around 10 minutes.

Then using National Rail Enquiries from the airport (code RIA) to Cardiff Queen Street I see £5-00 trains directly linking the two, at 06 and 12 minutes after the hour, either no changes or 1 or 2 changes as fast as 39 minutes (the 10:06 and 13:06 for example).

Would that help?

There's an even better route here, changing off the 304 onto Cardiff Bus 58, which reduces the walk to 4 minutes-
https://www.traveline.cymru/journey-planner-results/?search=1&typeaheadFromName=Cardiff+Airport+%2C+Cardiff+Airport+%2C+&typeaheadFromLat=51.399&typeaheadFromLng=-3.3399&typeaheadToName=CF24+3BA&typeaheadToLat=51.4869&typeaheadToLng=-3.16795&maxWalkDistanceMetres=1000&walkSpeed=4&from=Cardiff+Airport+%2C+Cardiff+Airport+%2C+&to=CF24+3BA&date=1677628800&timeMode=LeaveAfter&time=18%3A00&maxChanges=fewest&maxWalkDistanceMetresRestricted=

The 58 stops at Central Station so you may be faster to catch the train to Central Station, then the 58.
It also stops at Queen Street then 2 stops then the 4 minute walk- so the easiest route is train to Queen Street then bus or walk.

Using the Transport for Wales train site - put in From XCF (it says Cardiff Air Bus and this is the shuttle bus that takes you the short journey to the rail station at Rhoose (RIA) from which direct trains leave hourly for Cardiff Queen Street - so put To CDQ. You will see that the shuttle bus leaves the airport hourly at x51 to connect with the trains at Rhoose station. When the bus stops you will see 2 platforms - for Cardiff - you need the one furthest from the sea - which is clearly visible. The bus driver might ask for his fare separately and if so, you will need to buy a train ticket from Rhoose = £5. If you can't buy from machine on platform - the guard on the train will sell you the ticket. Note that on Sundays, trains are every 2 hours.
https://tfw.wales/purchasing-your-ticket/check-train-times-and-buy-tickets

Here is the timetable provided by the Adventure bus company that runs the 304 from the Airport to Cardiff - but the route is not direct as it goes via Barry:> https://www.adventuretravel.cymru/bus-services/cardiff/304/
You need to scroll to find the services starting at Llantwit Major station and you should get off at Custom House Street in the middle of Cardiff. The walk from here to St.Peter’s Street is about 10 minutes longer than it would be from Queen Street rail station from which it is I guess 15 minutes. Note on the timetable that these hourly 304 buses leave at different times on weekdays v Saturdays.

Well, you wouldn’t get all this on Rome to Rio - which most Brits have never heard of thank goodness!

I am primarily looking at taking the bus. I have tried working out routes on traveline.cymru but even when I find a route and save it as a favorite, when I go back to it all the info is wiped out.
I have three weeks of routes to plan and am overwhelmed. Im going to have to wing it and hope bus drivers are helpful.

Rome 2 Rio, more like Road to Ruin
This site comes up with some ridiculous suggestions. Ignore