First trip to Wales

I'd go to either Conwy/Llandudno or to Bangor/Caernarfon - direct Avanti train from London to Llandudno Junction (for the branch to Llandudno) or Bangor (short bus ride to Caernarfon).

The castles which immediately spring to mind are Penrhyn, Conwy, Caernarfon and Beaumaris all easily reachable from either destination,

At Llandudno you have the Great Orme to walk on, from Bangor/Caernarfon you could do any of the sections of the 130 mile long Anglesey Coastal Path- linking that with Beaumaris Castle.
https://www.visitanglesey.co.uk/en/about-anglesey/isle-of-anglesey-coastal-path/

From Caernarfon or Bangor especially there are lots of walking possibilities in Snowdonia using the Snowdon Sherpa Bus- https://snowdoninfo.com/sherpa-bus-service/

There is also Conwy Mountain. Have a look at this website for a map This one has pictures.

There is a great train that departs form the Conwy castle stop to Snowdonia. If you go to the visitor's center in the park they can give you maps and hike suggestions. It is beautiful! Plas Mawr in Conwy is totally worth it if you love a home built in the time of QEI and there have great docents full of info. Walk the town walls and go to the castle too. If you want to bypass the limited restaurant scene buy a meat pie or 2 at the butcher shop and a pint at the pub and picnic. Conwy is all walkable.

I split a stay between Conwy and Caernarfon a few years ago. There is a local bus that goes between the two towns. For a short stay, what I would do is base in Bangor. Both Conwy and Caernarfon are short bus rides away. (30 minutes to Caernarfon and just under 30 to Conwy.)

I traveled the area by public transport. The buses are used mostly by locals.