You might wish to consider staying in Cardiff at least 1 night as this will give you time to visit the National History Museum of Wales at St. Fagans as well as Caerphilly Castle = a 20 minute train journey to the north & also a Dr. Who location. (Staying in Cardiff will be cheaper than London). (You could combine Caerphilly Castle with Castell Coch by first taking a Stagecoach bus to Tongwynlais) .
www.museumwales.ac.uk/stfagans/ (free) (Bus 32 or 320).
Although you can book a through train ticket to Cardiff Bay, the journey is a bit complicated as you must change at Cardiff Central & Cardiff Queen Street & the Bay train station is about 13 minutes walk from Dr.Who. My advice would be to book in advance (for the lowest fares) the train from London Paddington (PAD) to Cardiff Central (CDF). Then, when you leave the train at Cardiff Central, go down the stairs but DO NOT follow the herd out through the main north entrance but go the other way in the subway to the south exit .
Then, keeping the railway to your left, walk straight ahead for 2 minutes to the bus stop on the FAR side of the road. Every 10 minutes or so, a number 6 (marked Cardiff Bay or Porth Teigwr) will come. Pay the driver for a return trip - which will be about £3.40 - and the driver will not give change! Get off at Dr.Who. http://www.doctorwhoexperience.com
On leaving Dr.Who, don't get on the bus but walk around the inner harbour passing the Norwegian Church to the Senedd = Welsh Government building. If you wish to see the modern architecture, you can go in for free. Then, head for the Wales Millennium Centre (Opera House), perhaps having a meal first in one of the restaurants on the harbour or in the WMC. (The WMC also has a Tourist Information Centre with free maps).
You could take the ferry out to view the barrage or up river to the Castle (instead of Bus 6). (For day trippers, the bus is the faster option).
Bus stop is by the WMC for the number 6 back to the city centre. Don't get off the bus when it reaches the station but stay on until at least the next stop. If you get off at this stop (Custom House Street), walk towards the modern John Lewis department store and then proceed N up The Hayes. On your right is modern architecture - you might like to walk into the modern shopping mall called St.David's 2. On the other side of The Hayes is Edwardian architecture - you might like to take a peep at the Royal & Morgan Arcades.
Straight ahead are some large trees with a street cafe. Continuing northward, you eventually reach Cardiff Castle and I would suggest that you would only have time for the cheaper self guided tour. If you want the dearer more detailed guided tour, it will take about 1 hour. www.cardiffcastle.com
On leaving the Castle, you may wish to visit The National Museum of Wales (closed Mondays) - which is next to the City Hall. However, if you wish to give this priority, you could stay on the number 6 Bus & ask the driver to tell you the nearest stop to the Museum - most days, it stops practically outside .http://www.museumwales.ac.uk/cardiff/
From the Museum or Castle you can wander back through the city centre shopping area to the Central Station. (Most streets are traffic free).
Find trains at www.nationalrail.co.uk but book ahead via https://www.gwr.com (Trains are usually cheaper on Saturdays - providing no major sporting event is taking place). A debit/credit card number is given which can be used at British stations to get your tickets from a machine (chip & pin needed) or a humanoid. If paying on the day, it might be cheaper to buy tickets for separate legs of journey with splits at Swindon & perhaps Didcot -
The train must stop at Didcot & Swindon for this to be valid. (All stop at Swindon but not all at Didcot - click 'Detail' to find out).