The train will get you as far as a bus will get you anyways. Either way you will be walking the same distance. By the looks of things the Eisteddfod event site is starting from The Flourish (the fountain area behind The Red Dragon Centre) so any vehicle will not get much further than Cardiff Bay train station.
By bus, you can get the 57/58 or the X1 from Pontprennau to City Centre, then the Baycar 6, 13, 25, 132 or X8 to County Hall or Cardiff Bay Station on the Lloyd George Avenue. You can also get the 1 or 8 services to Mermaid Quay. If you’re buying day tickets or returns, note that they’ll only be valid for the operator who issues them - for example, if you catch the X1 to town (a NAT group bus), you’ll only be able to use it on the X8 to the bay. Cardiff Bus or Stagecoach (the 25 and 132) will not accept them, as NAT won’t accept Cardiff Bus or Stagecoach tickets either.
I would also add that using a bus may well be cheaper than the train. A return on the train costs £3.40 from Cardiff Central, or £2.70 from Queen Street. Then there is getting to those stations as well. Cardiff Bus ‘Day to Go’ tickets are £3.80 for adults, £2.50 for children under 16, and can be used all day. If you are travelling all day with children you can get a Family Day to Go for £5.30 which covers up to 2 adults and 3 children (or 1adult and 4 children), on one ticket, but a minimum of 1adult AND 1 child must be present for the ticket to be valid. If you’re going with a group of up to 5 adults you can get a Group ticket for £10.00 - well worth it if you’re travelling with more than 3 people!
I don’t have any idea of NAT’s ticketing structure, but they probably have their rates on their website.
EDIT: as an addage, anyone travelling by bus from outside Cardiff and wanting to purchase just one ticket, the Network Rider is the only ticket you can buy that is valid on ANY bus in South Wales, regardless of operator. There may be restrictions on some services though (Traws Cymru services for example, but check before you travel).
If you’re coming in by train and not changing for the Cardiff Bay train, be sure to get the ’PLUSbus’ entitlement for the Cardiff area so you will be able to use your train ticket on the bus. The ticket must have ‘PLUSbus’ printed on the ticket to be valid on the bus.