Liverpool to Snowdonia National Park

the name now used is Eryri National Park since names in Welsh are used.

You can get there by a combination of trains, but not one train will take you the whole way.

Trains run from all Liverpool central stations to Chester where you can change for trains to Llandudno Junction. It is an easy journey!

There you can get another train which takes you down the Conwy Valley to North Llanrwst for Trefriw or Betws-y-Coed, both of which are on the eastern edge of Eryri | Snowdonia National Park and are the nearest 'entry points' and accommodation locations to Liverpool. (The park is not like a US national park in that there is no official entry / exit point for payment etc - it just covers a large geographic area!)

From either locations you can get buses further into the park to explore!

TFW website has times etc: https://tickets.trc.cymru/#/ - search for Liverpool -> North Llanrwst or Betws-y-Coed

What part of Snowdonia are you wanting to get too? It does cover quite a large area... If you are wanting to base yourself in Llanberis (terminus for the Snowdon Mountain Railway) you would need to catch a train to Bangor and then a bus.

Or you can also get to Llanberis from Betws-y-Coed Railway Station on the hourly S1 Snowdon Sherpa bus to Caernarfon.

I’m not sure what you’re saying isn31c. That I should go to Betws y coed and then take the Railway station there to Llanberis? What is in Llanberis? And then why the bus to Caernarfon?

Rather than going to Bangor for the Sherpa bus you can alternatively go to Betws-y-Coed on the train (especially if you were going there anyway). It was mentioned above as another entry point.
Then you take the bus bound for Caernarfon from Betws-y- Coed railway station as far as Llanberis.
Llanberis is a climbers mecca. As well as the train up Snowdon there is also the narrow gauge Llanberis Lake Railway, the National Slate Museum (closed in 2025 for redevelopment), and the remains of Dolbadarn Castle. Many people can spend a week there with the many walks, some flatter than others!
Caernarfon, with its famous Castle and harbour is just 25 minutes away from Llanberis on the bus.
You also have the Welsh Highland Steam Railway at Caernarfon to show you more of Snowdonia.
Many trips possible. Another one is to use the Sherpa buses from Llanberis to Beddgelert over Pen-y-pass (Beddgelert is on the Welsh Highland). Basically do a circuit of Snowdonia by bus.

Where you base yourselves will depend on what you want to do. If you want to do some walking in the NP then Betws-y-Coed or Llanberis would be excellent bases. If you want to tour and not necessarily do mountain walking then maybe Conwy (excellent castle) or Llandudno (seaside resort) would be alternatives bases. You could take a day tour of the area from there or hire a car and do it yourself.