The countryside is always open.... As for walking, a lot will depend on the conditions and drainage. After a wet winter, footpaths, especially over farmland could be very muddy. Forestry trails, particularly on forestry roads, should be fine for walking. Think about beddgelert Forest or Gwydir Forest Park, both in Gwynedd. Coed-y-Brenin is a bit further south. In South Wales there is Afan Forest Park. should be fine for walking. There is also Newborough National Nature Reserve and Forest on Anglesey. The sand dunes will be well drained and dryish under foot.
A lot depends on what you mean by 'destinations... It is still early for the tourists which is an advantage, however, many tourist attractions may still be shut, not opening until Easter. You will need to Google opening times to check.
You may be lucky with the weather, but early March could still be quite cold, especially allowing for wind chill. We've had many holidays in North Wales at the end of March and early April. Some have been warm and sunny. Others have been bitterly cold and we've needed all our layers - so make sure you come well prepared with plenty of warm clothes, including full waterproofs. (These are also good windproofs too.) We even experienced a blizzard on April on the Llyn Peninsula which was unheard of. Fortunately it didn't last long and the snow soon melted.
Where to go - that is difficult to say. Do you want to base yourself in a town? If so, think about Cardiff or possibly Llandudno/ Comwy area, or somewhere along the coast between the two. This should give plenty of choice of accommodation. Are you planning on hotel/B&B or self catering? YAgain you may find some places have yet to open for the season.