I take it that you are travelling from Newport in south east Wales. An express leaves Newport at 17.31 for north Wales and it arrives in Llandudno Junction at 20.53. The train continues to Holyhead but as it is an express, it does not stop at Conway. The next train from Llandudno Junction to Conway departs at 21.29 and takes 2 minutes to do the trip. So, you might like to use the express to Llandudno Junction (Fares can be pre-booked about 11 weeks in advance for about £29 to LLJ and this will include seat reservations). You may like to get a bus or taxi from Llandudno Junction (LLJ) to Conway.
Friday evening is the busiest time on the railways. If you go to www.nationalrail.co.uk and put in for NWP to CNW, you will get the timings for the journey - fares go up as date approaches. On the 17.31 from Newport, it will tell you to change at Chester to a slow stopping train to Conway = the 21.29 from LLJ. DON"T change at Chester but stay on the more comfortable express to LLJ.
Other trains also operate between Newport & Conway. If on a train to Conway, make sure the conductor knows that you wish to get off at Conway = a request stop. If on the express, it does have what they call Premier or Business Class for a supplementary fare. This entitles you to a meal in a dining car. The company that runs the trains on this route is Arriva Trains Wales.http://www.arrivatrainswales.co.uk/BusinessClass/
Please double check all facts & timings with Arriva.