HiI am from Chepstow...
The Severn crossing that is on the M48 (the junction closest to Chepstow it your best bet to cross on a push bike. The new bridge on the M4 is for cars etc only.
It has seperated paths on either side of the bridge for people to walk across, etc. Lots of people cycle across the old bridge and it is free for push bikes and pedestrians.
Be warned though, it is a really open estuary and so it gets really really windy on the bridge. And it is a fairly long suspension bridge, don't be put off if you find out that it wobbles just a bit. The view is good though.
The crossing is from Aust on the English side, to Chepstow on the Welsh side. There is no ferry between Weston and Cardiff.
If it is Weston direction you are coming from, you can get to the old bridge by going along the roads along the coast
I don't know about the youth hostels, but I think most hostels have a kitchen you can use, as that is usually the case in most other hostels.
If you don't fancy cycling across the bridge, catch a train to Bristol Templemeads, or Bristol Parkway and there is a train that will go from there to Cardiff - It uses the Severn Rail tunnel under the river, and the first stop is at a place called Severn tunnel junction. (absolutely nothing there, as it is just literally just a halt stop) or the second stop is Newport and then Cardiff.
I don't know how much it is for the train, it wouldnt be any more than £10 and anyless than about £4.00.
Any other questions, then I would be happy to help. Seeing as I am a local.
I hope you have fun anyway.
There's not a bad music scene around to be found too, have a look on a magazine web site, that concentrates on Wales folk...
www.folkwales.org.uk/taplas.htm
here's an address that has links for the Chepstow area and is the towns website. Some nice pics on here.
http://www.chepstow.co.uk/main-index.htm
But most of all have fun
Regards
Ella