I check replies to my own posts, then “Unread”, then “New”. With a liberal use of the “Dismiss” button.
Ding-ding-ding! Your method is the one I want to imitate. 
How do I do this? I’m looking at my profile page. I do not see an option like that under “preferences.” Where do I find it?
Is this “unread” on the main page or “unread” under the hamburger menu? Or is that the same place?
I easily see in the dropdown under my avatar picture threads I’m watching, as well as threads where someone quoted, replied to, or @'d me. You seem to be saying I can identify these threads/posts from the “unread” page? I’m not following…
On your Preferences page select “+Tracking” from the menu across the top.
In the top section labeled “Topics” select [created in the last week], [after 4 minutes], and [Tracking]. Scroll down to the bottom and and click [Save Changes].
If you want to completely ignore any categories, select them in the “Muted” control just above the [Save Changes] button then click [Save Changes].
Same place. The URL is https://boards.straightdope.com/unread
Go to the unread page as above. You’ll see a long list of unread threads. Now click your icon at upper right. Any thread where you’ve been @ed, replied, or quoted, but have not yet looked at the post where you were @ed, replied, or quoted will be highlighted differently from the rest. Each theme uses different colors, so I can’t say what color they are on your screen. But you can tell which are which.
As an example, right now under my icon I see a reply to me by wolfpup in the thread “Random “Why don’t we have this in the US” thread”.
So now I look down the list of threads in the main body of the page looking for the thread whose title is “Random “Why don’t we have this in the US” thread”. Then I click that.
The reason to do this two-step process is if I click the highlight under my icon, that takes me to where wolfpup quoted me. Which is nice, but if there were 15 posts between where I last read and where he quoted me, I’ll skip all those. Oops. By clicking the thread title from the main unread list it takes me to the first post I’ve never read. I can then read downwards from there and eventually encounter wherever wolfpup replied.
My overarching purpose of my whole technique is to
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Prioritize replying promptly in those threads where I’m part of an active dialog.
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Once I’m caught up in those, I want to catch up the threads I’m following, and may have posted to many times, but where I’m not now immediately in the flow of the dialog.
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Next, I want to add any interesting new threads I’ve never seen to those I’m already paying attention to.
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And last of all, I want to take the new threads I have no interest in, and “move them out of my in-box” so I won’t be bugged about them again.
At least until somebody in one of those threads happens to say something like "Hey, I bet LSLGuy might have something useful to say. At which point I’ll be summoned to the thread.
the what? We have a dismiss button? What does it do?
I agree with your objectives. Thanks for the detailed instructions. I don’t have time to work through them at the moment, but I’ll get back to them. 
There’s a [Dismiss] on the “New” page that makes those threads “un-new” for you. So they disappear from the “New” page, but are still visible on the relevant category page, the latest page, etc.
There’s a [Dismiss …] button on the “Unread” page which has two functions. The first one is to declare all those threads to be fully read to their current bottom. In effect it’s as if you went into each one, read to the bottom, then closed the tab. They’re still being tracked by you and will reappear in [unread] once another new post occurs. But not until.
There’s a checkbox in the popup you can check which says “stop tracking these topics…”. If you check that before clicking the [Dismiss] button, then their tracking state is reset to “normal”. Which means they’ll never show up in unread again even when new content is posted. They’ll still be in their category and in Latest.
The key thing for users to understand overall is that the various category pages and the "Latest" page are the same for everyone, but New and Unread are personalized for each of us based on which threads we've read, how our profile is set up, etc.
Once you learn to drive the system so new and unread feed you what you want, it’s really a pretty nifty system.
It’s at the top and bottom of the Unread and New pages and clears the queue.
I start with Suggested Topics, which includes Unread, and often that’s all do. Maybe once every couple of weeks I go to New.
The definition of “New” is adjustable under your profile → Preferences → +Tracking. Depending on which of the 6 choices you’ve selected, the New page may or may not show you everything new since you last looked. That might not matter to you, but I bet some folks will be surprised that “New” isn’t necessarily “Everything New”.
I just open up at categories and it has every current thread in order except for those in the 5 categories that I have muted. As I lose interest in a thread - perhaps the discussion has become tedious, I have learned what I wanted from the thread, I don’t care about the question or the subject, I mute the thread. At the moment I have muted everything back to March 2023 except the few current threads that I remain interested in. Typically, unless someone chooses to engage with me in this thread, it will be muted in a day or two.
Same, only it goes ATMB, FQ, GD, CS, IMHO, BBQ. Every second day or so, I’ll add MPSIMS, GR and PE to that rotation. Of course, if I have any notifications, I go to those first.
I’m looking over your full instructions this morning…
Wow --I had NO idea so much fine-tuning was available under “Tracking.” That is very helpful.
Question: under “Notifications” the option says, “Notify when liked [with some choices.]” But we don’t have likes set up, do we? So this option doesn’t apply?
Right under that is “Live Notifications.” Mine says “permission Denied,” apparently because of my browser settings. But in fact, I do get notifications pretty quickly-- a tiny number appears superimposed over my avatar in the upper right corner. So am I missing anything? (I have a bad case of FOMO-- hate to miss out. 
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I will be exploring further. Thank you so much. Ignorance fought!