There's not loads in Fishguard itself. You can find a nice picnic spot and watch the ferry, there's a swimming pool at the leisure centre and a few parks dotted around.
A short drive away you have far more...
Castell Henllys Iron Age Fort,
Hangar 5 trampoline park in Haverfordwest,
There are loads of beaches nearby along the Pembrokeshire coast or even head over to Ceredigion,
Picton Manor,
Scolton House,
Clerkenhill,
Dyfed Shire Horse Farm,
The bug farm in St Davids,
St Davids Cathedral
Oakwood (can use Tesco vouchers). It was looking a bit run down when I was there last but the boys really enjoyed themselves).
I think there's a chocolate place in Haverfordwest...I've just googled and it's the Wickedly Welsh Chocolate Co.
A bit of a further drive you have:
Folly Farm (can use Tesco vouchers if you have them) - my eldest is 6 and loves it there,
Pembroke Castle,
Cardigan Island Farm Park (if you head to Cardigan the Cardigan Bay Brownies are lovely as is the Pizza Tipi),
The Wildlife Centre in Cilgerran,
Heatherton World of Adventure,
Manor Park,
The Dinosaur Park.
As long as you're prepared to drive you should be able to find plenty to keep a 6 year old very happy!