Day trip to Cardiff with a 5 yr old

I'd keep up with the plan to go to St Fagans. A Taxi will cost around £10 but entry to the museum is free. Ask the taxi to drop you off by the church in St Fagans and enter via the back entrance. The gate may look closed but it's just a buzzer entry system and they will open the pedestrian gate for you. You'll start at the castle side (really a stately home) and you can view some but not not many of the rooms. There are lovely gardens to relax in and explore but you need to make you way down to the other side of the lake in order to find the museum 'proper'.
If you decide to stay in the city centre then a trip to Cardiff Castle is probably your best bet. Then take the aqua bus (hourly schedule) down to Mermaid Quay which is Cardiff Bay. Plenty of food options here and usually something going on during the summer. If she's worn out at the end of all that just take a taxi back up to Central station, it's a short journey.

I thought Cardiff Bay looked really kid-friendly. As a lone adult I didn't find it very interesting when compared to the center of the city.

Cardiff City centre is compact and right by the Central Station. Cardiff Castle is about a 15 minute walk from the Central Station so that is a possibility. If having been into the Castle (for which you have to pay - though you can see the outside for free) - you could leave the main gate & walk west alongside the ‘animal wall’ (stone animals) for about 6 minutes to enter Bute Park by the Pettigrew Tearooms (worth a stop for a break) prior to entering the park. A short walk on the path going north and veering left will bring you to an extensive flower border before reaching another cafe. However, near the Pettigrew tearoom, you will notice a pontoon for ferries to the Bay = not a beach but a huge harbour with yachts. So, once having seen the flowers, you could take the ferry down the river to the Bay to land at Mermaid Quay - this takes about 25 minutes. On Mermaid Quay, you will find loads of restaurants and nearby is the Wales Millennium Centre - a concert hall/opera house as well as the Send (Welsh Government Building) and the Norwegian Church. If you go in the big red brick building and enter first room on left, press a button and a short video will play telling you about the history of the place. https://www.cardiffcastle.com
http://www.mermaidquay.co.uk

To return to the city centre - take bus 6 from outside the WMC and ask the driver to tell you the stop for the Central Station - of intending to go straight back on the train. Alternatively, get off at the next stop and wander around the city centre - which is a mixture of modern malls and Victorian arcades. Please note that the bus driver will not give change - just ask for a one way ticket to the city centre. (It should cost about £1 or £2).http://www.visitcardiff.com

Regarding St.Fagans: well you can’t walk it from the city centre - you would need to get the 32A bus - which is every 30 minutes usually and is run by Easyway. The journey takes about 24 minutes. Entry to the St.Fagans site is free and you would enter first the western part of the site rather than from the Castle end. The 2 halves are linked with a small tunnel. At the entrance building, you will find a restaurant and just outside is a kids play area. If you are going to St.Fagans, I would do that first as it takes about half a day to wander around & see everything.

PLEASE NOTE: in an earlier post, you said that you intended to do Cardiff on Thursday July 5th. From 16th June to 9th July, the Seven Tunnel is closed for electrification. This is the direct line from Bath to Cardiff passing under the estuary. During this time, passengers will be taken by train between Bath & Bristol Parkway where they will be transferred to buses for the journey over the Severn Bridge to Newport. Here, they take the next train that is going to Cardiff (pretty frequent). This means that instead of taking just over 1 hour, the Bath to Cardiff journey becomes around 2 hours. Go to www.nationalrail.go.uk - put in your date and joinery and you will see what I mean.

Oh I had no idea about the rail replacement...thats far longer than I wanted to spend on the train and/or buses. I am hoping to find a fun place for us to visit by train that is around a 1 hour direct ride from Bath. Any other suggestions on places that are a direct train ride from Bath and not too long a train ride (ideally under an hour?)

Yes, you could take the train going SE from Bath for about 55 minutes to Salisbury. When you arrive at Salisbury station, ask someone where the bus stop is for the Stonehenge shuttle bus:>http://www.thestonehengetour.info/timetable
https://www.salisburycathedral.org.uk
http://www.english-heritage.org.uk/visit/places/stonehenge/

Take the bus the relatively short distance up to Stonehenge. (Old Sarum is really an Iron Age hill fort = just a series of hill top ridges - so you might wish to skip that). Anyway, after Stonehenge, return to Salisbury and have a look around the medieval cathedral - one of England’s finest. Be sure you know the times of the trains back to Bath. There is about 2 per hour.