You're very much in luck. It's more how much time do you have? You're smack in the middle of Pembrokeshire, one of the best places in the UK for family holidays.
You're within a few miles of many of the UKs finest beaches, my favourites are Broadhaven South, Barafundle, Newgale, Tenby South. Haverfordwest is a stone's throw from Broadhaven North. Basically all are miles of sandy beach, beautiful views. Don't think that because it's February you shouldn't visit them. We were on Saundersfoot beach on Xmas Day. You'll have them to yourselves. Depends how traditional seaside resort or unspoilt and empty you want.
You're also near the Pembrokeshire Coastal Path for walking, and there are plenty of bits you can do with a kid if you don't mind carrying. There are guides online or I'll link when not on a phone. A good family day is Stackpole Lily Ponds, Broadhaven South Beach and lunch at the old cafe in someone's house in Bosherston. Tenby is a very kid friendly town with a soft play, you can go on a seal safari and visit the monks and chocolate on Caldey, which also has lovely walks.
Manor House Park is the zoo taken over by Anna Ryder Richardson with loads of space for the animals and a proper breeding programme, you can go in with the lemurs and wallabies and there's lots of really endangered species with lots of space. a lot of local parents buy the cheap season ticket and use it as a huge outdoor soft play in summer. There's loads of balance bikes and toy tractors for the kids and swings etc. The cafe is always rammed with families. It's near St Florence. It's open, we went over Christmas.
Folly Farm has a more traditional but huge zoo area - also an absolutely vast indoor soft play, an indoor fairground, a petting farm and outdoor rides. Every South Welsh small child probably comes here once. A good rainy day option. It's between Narberth and Saundersfoot. It'll be open.
You could also drive up from Haverfordwest to St David's, stop at Newgale pub have a Potter on the beach, up to Solva a beautiful little village worth a look around, go around the cathedral and town at St David's and back.
The whole area is geared up for holidaying families, and even out of season there'll be stuff on. a'm not sure specifically about Haverfordwest but I have a work colleague from there I could ask. My partners parents live down there and we spend half the summer there, we love it. Happy to have pms!
