One Week Wales Itinerary Late June

For the north, I'd recommend Conwy for a base as it's easy to get to with public transportation but close enough to explore Snowdonia (Eryri National Park). It's very well connected and should give you some guides to choose from in the area who can take you on a tour of Snowdonia (it's very hard to explore that area without a car).

Exploring the island of Anglesey is well worth the effort (not far from Conwy) as the far end of the island is where you'll find Holyhead, Beaumaris is well worth a stop among other sites on the island.

I would let others comment on south. We stayed in St. Davids which we loved, but it's quite remote and harder to get to without a car.

Are you coming off a transatlantic flight and are you likely to be jet lagged? If so you need a day to aclimatise

With just a week don't be too ambitious. I would suggest 2-3 days in Cardiff. There's lots to see and do in the city. Don't miss Cardiff Castle. Spend a day at St Fagans, the folk museum a short didtance from Cardiff and reachable by bus. This has over 40 old building from across Wales that have been reassembled on the site. There are rural farms as well as town buildings and it uis a real blast of the past. .

Caerphilly with its ruined castle is easily done on the train from Cardiff and there is also the fairy tale Castell Coch which is a short bus ride from Cardiff.

Then take the train (it's about a 4 hour journey) to either Conwy (walled Medieval town with a ruined castle) or Llandudno (Victorian seaside town with a prom and pier) for the other few days. The two are close and connected by bus. Llandudno has more choice of Accommodation.

Spend a day You can easily spend a day exploring Conwy (and don't miss Plas Mawr, the restored Elizabethan Town House in the centre of the town, walk the walls etc. ). From Llandudno there is the Victorian Tramway to the top of The Great Orme. The Bronze Age copper mines on the Great Orme are well worth a visit too. (and you can go into the old workings) By bus you can visit Bodnant Gardens.

Catch the train on the Conwy Valley Line to Betws y Coed to catch the Sherpa Bus S1 which will take you through the best of Snowdonia to Caernaron and then catch a bus back to Conwy or Llanberis.

It is then just a 45 minute train journey to Holyhead.

Once you have plans fleshed out come back for more information on public transport etc...

You have received some good advice so far. My wife and I spent four days in north Wales last May, based in Llandudno, and enjoyed it a lot. While undoubtedly Snowdonia can be reached by public transportation as suggested, we found it much more efficient to do a day tour that included Snowdonia, Conwy, the Great Orme, Carnafon and some other places. The tour was called Snowdonia and 3 Castles, operated by Adventure Tours Snowdonia. Note that if you google this the highlighted sites will be third parties such as Viator and Get Your Guide reselling this tour, but the company also has its own UK website where we booked directly. We liked Conwy a lot but thought that the larger Llandudno was a better base of operations to stay in. I second the recommendation of Bodnant Garden, which we visited by bus, and June will be a great time to go, but check the bus schedule carefully online for the timing since service was not frequent on that route, only about every two hours. Buses between Llandudno and Conwy are very frequent, however.

You have been given sound advice so far. You don't say where you are travelling from but I suggest you check out the air fares from your start point for doing this trip to Cardiff first and then Ireland v doing it in reverse.

Cardiff airport is about 10 miles SW of the city centre and the public transport links are not that good. Outside the terminal building you will find an Adventure Bus 905 that leaves hourly at x35 minutes for the 10 minute run to Rhoose Railway station. If using this, you need platform 1 for trains to Cardiff (which might have Rhymney as final destination). Trains on P2 on the seaward side (you will see the sea) are going in the other direction. The trains to Cardiff depart P1 hourly at x04 - except Sundays when it is every other hour. From Rhoose station to the city centre takes 30 minutes.

You may also see a First Bus 304 outside the terminal building. These are also hourly but note that they take about 1 hour 30 minutes to reach the city centre via a wiggly route. OK if heading to Cardiff Bay but otherwise the former is the better way to go. Also consider sharing a taxi with someone else (to split the fare) if heading into Cardiff.

You might like to consider searching AirBnB for small holiday lets but some of them charge more than chain hotels such as Travelodge and Premier Inn. Be sure to read reviews from clients. I did a quick search and this AirBnB came up 2 miles west of the city centre with buses nearby every few minutes. I have not stayed in this place but just pointing out something that might be what you are looking for? (The Cardiff Bus 32 to St.Fagans stops 5 minutes walk away).

https://www.toursofwales.co.uk is a company that run small group tours. (Not tried them as I use a car).

For north Wales - suggest you consider Conwy or Llandudno. Direct trains go from Cardiff to Llandudno Junction every other hour & here you must change for the short hop into either place - or use a bus. It is cheaper to pre-book a specific train in advance - but not too far in advance as the cheaper fares might not be loaded. Expect to pay around £31 to £43 for the one way trip (without Railcard).
I find this site is easy to use and it gives some idea of when the lower fares are loaded - though the very cheapest prices are at inconvenient times. https://www.buytickets.scotrail.co.uk

You could buy a combined train and ferry ticket for £45.70 (this price does not vary). A sample journey would be leave Llandudno Junction at 11.50am ; arrive Holyhead at 12.36; ship departs at 13.15 and arrives Dublin Ferryport at 15.45. (Some trains to Holyhead also call at Conwy).

A well as Conwy and Llandudno it’s worth having a look at Deganwy for accommodation. It’s right next to Llandudno so you might already be getting search results for it but it’s worth a specific search.

Rather than taking the ferry from Holyhead have you looked at flights from Liverpool or Manchester? They are both easy to get to from North Wales and have regular flights to Ireland.