(I should preface this by mentioning that my browsing is 99% website based, hardly anything by phone.)
I do all of the above, but what I actually do most is re-visit saved Identify bookmarks, for either places or people. For example, a long-term project of mine (and several collaborators) is revising the vascular flora of the White Mountains in California and Nevada. So my bookmark for that place (developed using Filters in the Identify view) is:
An example of a bookmark for a specific user (again developed via Filters) is:
https://www.inaturalist.org/observations/identify?quality_grade=needs_id%2Cresearch&user_id=jdjohnson&taxon_id=211194
Each time a bookmark is revisited, all newly posted observations meeting the same filter criteria show up at the front of the line (by default – this can also be changed in the Filters).
I prefer the Identify view to the Explore grid view because it initially looks similar, but comes with a lot more functionality for efficient browsing of, and input on, the observations. It took me a while to discover and learn that functionality, but now I can’t browse without it! But if one prefers the Explore view, just remove /identify from the URL - most of the filter options are the same in each.
You may notice that I include “Needs ID” and “Research Grade” observations in my filters. This is not the default filter behavior in Identify mode, and I need to specify this to make sure I see everything, and catch any mis-identified Research Grade observations. (Also, my taxon filter for each is for vascular plants [Tracheophyta], since I am not much use with anything else. 
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Because there are so many different browsing preferences for the vast diversity of iNaturalist users, I have often wished that some of the Filters defaults could be changed and saved as “sticky” account settings by each user. There is an open feature request for that. There is also a request to save bookmarks internally instead of in one’s Internet browser (or, like me, in a separate text file).