Why should evolution not be equated with progress?
I'm glad you've asked the question as it is a common layman misunderstanding.

Dog vs Human

Your example comparing humans and dogs is actually very central to the logical flaw that yields many to equate evolution with progress.

There are several issues when you say


  humans do seem to be more advanced than dogs



Humans are not more evolved than dogs


Humans, dogs, jellyfish, oak tree, fungi, bacteria, .... we have all evolved during about 3.5 billion years. None of the extant lineages is more evolved than any other at least in terms of the number of years of evolution. Note, however, that what matters most when considering evolutionary time is the number of generations. Humans having a rather long generation time, it tends to make fewer generations. In this regard, one could expect a dog to have been through more generations than humans and could eventually be called more evolved but even there it is not that easy. See also Are we “more evolved” than present-day bacteria?

You are a human, be aware of your biases


As @Jamesqf rightly said in the comments, you did not feel like considering the dog's extraordinary sense of smell, better hearing, higher jumps, stronger jaws or protective fur. Not to mention, dogs can typically raise way more offsprings than humans can. Not to become personal but my 8 months old, 9 kg Cocker Spaniel runs 300 meters at the same time Usain Bolt runs 200 meters! You seem to have only considered things that make you human such as opposable thumbs and high cognitive abilities. Coming to cognitive abilities you will note that a puppy learns by association faster than a baby human. Dogs just plateau much quicker than humans.
Make sure that you understand that high cognitive abilities are not a goal or a direction of evolution. While it is hard to measure the evolution of intelligence for both a question of definition and question of measurement tools, it is very very likely that many lineages (incl. eventually the Homo lineage) have evolved toward reducing intelligence at some point (see this post).
You say humans could wipe out dogs from Earth however again it feels very human (and awkwardly combative and hostile) to consider the ability to kill as a measure of progress.

The illusion of considering two variables


You consider a correlation between two variables 1) how evolved a lineage is and 2) how advanced a lineage is. (As discussed in the first point, there is actually no variance in how evolved lineages are if we consider evolutionary time). You are determining those that are more evolved by how "advanced" you visualize the lineage. Your explanatory and response variables are therefore the same and you do not explain anything by saying 'humans are both more advanced and more evolved than dogs so evolution should equate progress'



Definition of progress

A big issue in your question is the definition of progress. For example, if you define progress as an increase of mean fitness (the concept of fitness in biology has nothing to do with bodybuilding, see here) over time, then yes evolution is in part to be equated with progress as increase in mean fitness over time is exactly what natural selection is doing (see the so-called Fundamental Theorem of Natural Selection).

Evolution ≠ Natural Selection

So, let's assume we define progress as an increase in mean fitness over time. As stated in the previous section, this is exactly what natural selection is doing. So, instead of saying Evolution should not be equated with progress, one could say Natural selection is to be equated with progress (as a reminder this assumes a very specific definition of progress)

You will note that I replace the term evolution with natural selection. It is often unclear to laymen but these two terms, while related, are different. 

Evolution is a change of allele (loosely speaking, an allele is a variant of a gene) frequency over time. There are a number of mechanisms that can cause evolution, one of which is natural selection (which itself is caused by a fitness differential associated with genetic variance). But there is way more to evolution than just natural selection. Genetic drift, mutations, migration and other demographic elements are all important factors causing the evolution of populations.

You should have a look at Understanding Evolution by UC Berkeley. It is a very introductory online course on evolutionary biology that will teach you about the basic forces that cause evolution.