How can I neaten up the corner between stairs and wall
There are 2 typical ways to neaten things like this up:


Trim: thin pieces of wood or plastic cut into short pieces that fit in the treads and risers, tight to the wall. These can be painted or if it is wood, varnished. Pros: looks good, Cons: material cost, requires tools and expertise to look good, installation time
Caulk/mastic/silicone bead: gel sealants which are squeezed from a long tube in a continuous line that would trace the contour of the stairs between the stair and the wall. Pros: looks decent, cheap, fast and requires few tools. Cons: Not as nice looking as trim, only covers the crack between the tread and the stair, does not cover up the wall above the tread


You have a challenging set of stairs to trim because the front of the stair is curved, so any trim would need to match that contour exactly in order to look good. This can be done with the right tools, it's just a matter of expertise. Some sort of sealant bead would be a much simpler and cheaper solution, I would suggest paying someone to do it as they will do a better job of it. 

Note: the above assumes you want to keep the look of the wooden stairs, if not you could carpet them up to the wall. If you do plan to carpet them nothing else is necessary.