More damage

Something was hungry...

BTW, if you are tempted to round off that stair tread, you can also use a palm sander with coarse grit paper. It will take longer, but you are much less likely to mess up.

Replace the door trim and bottom step.

Sand and refinish the post.

Replacing the door trim/step is not an option.

I have started to fill the gap in the door trim with wood filler.

Also the newel post has been filled. The animals did a huge amount of damage to it! I am astounded at people who let their cats use the house as a scratching post.

That looks like the work of a dog to me, not a cat. I would also try to get some replacement pieces just right for the the trim and step, but it might be hard to get that step, in particular, to match. I guess if what you're doing doesn't turn out very well, you could backtrack and replace since you are doing this yourself.

Good grief! That is really something.

How historic or old is your house? Being somewhat lazy, and also into preserving original wood wherever possible, and because finding new wood to match aged historic wood can sometimes be difficult - I've had success with repairing trim & furniture with a good wood fill product - something that can be shaped and sanded. Theres the elmer's water based brand which is good for smaller gouges and missing pieces,and Abatron which I think is epoxy based which is good for things that have to be built out more. Then, if you have a good eye for color, recreate grain using artist oil colors. When the finish coat goes on over the whole piece you don't notice the difference unless you look closely. Obviously this is an approach that would work better on things not at eye level or that you dont look at really closely - trim near the floor, etc. Ive been using it to repair sashes that had rot in some places.

Well, its a thought - just wanted to offer an alternative.

Definitely dog damage.

Not sure why replacement isn't an option (it's by far the easiest way), but if you need to preserve as much as possible of the original wood you can make a dutchman (lots of examples on google and youtube) to fill in the missing wood using a well-matched piece of scrap and a bit of artistry (cheap old furniture is often a good source for bits of hardwood or old-growth softwoods). Unless you get really good at faux-finish painting a wood filler patch will probably look worse than just smoothing off the rough edges and leaving them as they are. For the stair tread I'd either replace it or rip off the front inch or two and glue a new strip in place. To make it less visible you could add the replacement strip to the back edge. It's important to keep stair treads the same width, a half inch difference can create a fall hazard.

As a follow-up to this thread five years later, I ended up just filling it with wood filler and then I stained it to match the existing trim with a little bit of faux graining I think as well and it turned out great. You could not really tell unless you got up really close with your nose to the wood. I quite enjoyed undoing the damage that those animals inflicted.