When people talk of the north–south divide in Wales, it's not just about language – part of it is physical. The barrier created by the Cambrian Mountains, Brecon Beacons and Snowdonia means that it's often quicker to duck in and out of England to get between north and south Wales. The same is true by train: there's a network of lines that slowly zigzag their way through the country but the faster trains head through Bristol and Birmingham. That said, both roads and rail lines are extremely scenic. In Wales that old adage about the journey outweighing the destination is aptly demonstrated.
Wales is one of those places where Brits come to get back to nature, so it's extremely well set up for walkers and cyclists. With a flexible schedule and a modicum of patience it's quite possible to explore the country by public transport. However, it's worth considering hiring a car for at least part of your trip, especially if you're on a limited time frame and you're not averse to losing yourself in the sort of narrow country lanes that require pulling over when a car approaches from the other direction.
Buses are nearly always the cheapest way to get around but you'll generally get to places quicker by train. For information on services your best bet is the local tourist office where you'll be able to pick up maps and timetables. For up-to-date information and a journey planner covering public transport throughout Wales, visit Traveline Cymru.
If you're planning a whirlwind tour of Wales by public transport, you might like to consider an Explore Wales pass (www.arrivatrainswales.co.uk/ExploreWales/; adult/child £99/50). It allows free travel in Wales and adjacent areas of England on all rail routes and nearly all bus routes. The passes allow unlimited bus travel plus four days of train travel within an eight-day period. Cheaper passes (£69/35) are available if you're only wanting to visit South Wales or North and Mid-Wales. The passes can be bought at most staffed train stations and rail-accredited travel agencies in Wales.