Hello Angie,
I am not local to New Cumnock, but I was there last week.
Turn right out of the station and head south towards the centre on the A76 main road. After just over half a mile, the road turns sharp left, but take the right fork (B741). After a few yards up a short hill, there is a left turn (Afton Road) marked "Cemetery" and "Burns Memorial". Follow this residential street for another half mile in a southerly direction and the cemeteries are just after the brow of the hill.
The new cemetery is first with the old one adjacent. There are lots of paths in both but there will be some walking on grass to read stones. Both places are very well looked after and almost all stones are legible.
If you are ever in Old Cumnock, the Baird Institute has books of memorial inscriptions up to 1918.
We did not stop to eat or shop in New Cumnock but there did not appear to be much in the way of facilities.
Regards,
John