It helps that at this point I have a lot of subconscious habits about how I arrange tabs automatically. For instance, when browsing something like HN from the front page, if a link interests me I'll middle-click the comments link first, opening it in a sub-tab. If I'm interested in the article as well, then from the comments sub-tab I'll middle-click the article link, generating a sub-sub-tab. I don't even think about this; it's just how I've browsed sites like HN for a long time at this point. I end up with a nice little structured recreation of whatever part of the homepage I'm interested in at that moment. Then, if I don't want to worry about HN right now, I just collapse the entire tree and come back to it when I want a break.
As far as other keyboard shortcuts, I would probably use TST more if it offered "search and goto tab" within the TST window itself, including tab-key to fold/unfold sub-tabs, and return to select the actual tab.
I wonder if I've been trained by Emacs not to need to visualize these things. When I was first using it, I definitely felt a little uneasy without a visual "overview".