How to round the edges of this object? I need to round the edges of this object.

The bottom edges are going to be rounded in conjunction with another object, so, I guess, this should be left out right now.

I tried using the bevel tool (and a bevel modifier) on an edge (series of Blender edges), but I got the problems you see on the images. When selecting all edges, the bevel doesn't know any limit and isn't applied symmetrically.

When selecting only the edges shown (or only the other ones), I can't proceed with the rest of the bevel, and strange little artifacts appear.

FYI: The top surface is derived from an icosphere.



Without the subdivision shown above, a stair comes into existence (bad):



Whole object:
Unfortunately I believe you have started on the wrong foot.

You object has a very bad topology, it's not quad based, it is full of triangles. very unevenly subdivided with extremely high density on top contrasting with huge ngons on the sides, and unnecessarily subdivided in width direction.

All these things make harder to work with, Blender operators will misbehave, that is why you encounter these problems and are likely to expect further down the road.

Modelling applications generally like four sided faces, my advice is to start over with a better mesh base and work from there.



Watch some beginner tutorials on hard-edge modeling and follow these guides for topology