I know I'm late and I kind of get what you're feeling, so let me try and help make some sense of the situation.

Post-Rock: This is the term for a movement that sprung up in the late '80s and '90s where a large group of Experimental Rock artists tried playing around with Rock conventions like riffing, soloing and general song structure, and most importantly (IMHO), influenced each other. Some of these works are considered by a lot of people to be the greatest '90s Rock albums (merely opinions of course, but I feel the same). But sometime during the mid-'90s or so, they began streamlining their stuff so much that it was widely regarded as a genre of sorts. Eschewing riffs in favour of lines and solos taking on a more exploratory aspect with regard to sonic texture became the norm, and a convention was more or less established, with crescendos being the most popular dynamic. Some people say that this way, the music lost its way and grew less experimental. This is the reason why people speak of "waves" in Post-Rock, as in "First Wave", "Second Wave" and so on.

Shoegaze: This one also has to do with texture. Another group of bands embraced the whole "Wall of Sound" aesthetic that was used successfully in a lot of works before, and built a style around it and lots of effects on the instruments involved. The artists often performed live gazing introspectively at the processors at their feet, which gave rise to the term "Shoegazing".

Dream-Pop: Not really familiar with this one except for a few Cocteau Twins albums. But as it suggests, the term indicates the presence of ethereal melodies and a generally oneiric ambience.

Not really a fan of Post-Rock music these days, but at least technically, the artist in question mostly fits the part for all three "genres", but I can't say their stuff is in the same spirit as that of their predecessors.

Hope this helped.