Yes, you can get from the Cotswolds to Conway by mainly driving on busy multi lane motorways - that take you through the Birmingham conurbation. However, if you read my response to 'urzababa' (below), you will find a route on slower roads that take in interesting places along the way. This route of mine goes to Conway via Llangollen.
Both Llangollen & Conway are nice enough places. What you must realise that to see north Wales, does not just involve driving to these towns and then ticking off that you have 'done' north Wales. To see north Wales proper, I would suggest that you should consider visiting the following Castles > Harlech, Caernarvon & Conway. The Italianate village of Portmeirion (near Porthmadog) and take a trip up Snowdon (by mountain railway from Llanberis) - if the weather is good. Bodnant Gardens is also well worth a visit. Then consider also visiting the coastal areas - but not that which is east of Llandudno - which is not good!
All Americans seem to be heading for Conway because of Rick's advice. For those using public transport, this is indeed sound advice but for travellers with cars, consider a much wider area and also check that any place you plan to stay has free parking. You may like to consider staying on a farm.http://www.farmstay.co.uk/PlacesToVisit/Wales
Always check on Google Earth to see exactly where intended hotels, etc. are located. If staying in Llangollen, you are further away from the main attractions further west - though the area around Llangollen is indeed very scenic. The drive from Llangollen to Barmouth via Bala is very scenic. You can cover the main attractions of north Wales from either Conway or Llangollen but Conway would have the edge in terms of distance from places like Snowdon, Caernarvon & Portmeirion. You are unlikely to find free & easy parking in either town.
PS. If you want a really scary walk in the Llangollen area - walk the Pontcysyllte Aquaduct http://www.pontcysyllte-aqueduct.co.uk;