I’m not entirely sure what you mean by ‘each season is independent’ - if you mean that the story arcs don’t carry over seasons, that’s more or less true, although they frequently set up upcoming arcs in earlier stories.
But, until the new series, season-long story arcs weren’t the norm - there was the Trail of a Time Lord (the last story featuring the Sixth Doctor), and eventually the Seven Doctor stories started building Something Big…but the series ended before they could do anything with it. Beyond that, the series is mostly standalone stories, even when there is an overarching arc - although the 2010 series incorporated the larger arc into the otherwise standalone stories much more than was done, say, with the 2005 series arc.
And the Doctor changes at random intervals, dictated mostly by real world concerns.
Mostly, the actors just thought it was time to move on, but a few cases otherwise:
William Hartnell (who played the First Doctor from 1963-1966) retired due to ill health.
Colin Baker (The Sixth Doctor, 1984-1986) was fired, at BBC’s request.
Sylvester McCoy (Seventh, 1987-1989) was replaced because the movie was meant as a fresh start.
Paul McGann (Eighth, 1996) was the same, only replace ‘movie’ with ‘new series’.
Christopher Eccleston (Ninth, 2005) had only ever intended to do one season.
Aside from Eccleston and McGann, every Doctor has appeared for at least 2 seasons - Smith’s Eleventh is currently on his second, and Tennant’s Tenth had 4 seasons (3 regular ones, then a brief hiatus, and the Season of Specials that lead into Smith). 2 and 3 is fairly common for the earlier series, aside from Tom Baker, who was an outlier at ~10 years. In the new series there’s a bit more turnover for Companions than there is for Doctors - Rose and Amy and Rory have all done 2 seasons as regular Companions (Rose with two separate Doctors), but Martha and Donna only had one season each, Jack, Mickey, and a couple others had only sporadic appearances, and the Season of Specials didn’t have any recurring Companions at all.
As to what to watch next…the only relatively consistent answer you’ll get to that is ‘the entirety of the current (2005-present) series’ - where to start on the older ones is a bit of a debate. My recommendation for that would be to start with the 1970 season (Jon Pertwee’s first as the Third Doctor)…mostly because that’s the first run that the majority of the stories are still intact. (Note that unlike the current series, the original series was done, mostly, as serials - mostly 2-4 half hour episodes each story, though there were a few that ran longer.) Also, you might want to give Torchwood a look (after watching the 2005 and 2006 series)…reviews of that one are mixed, but I find it mostly enjoyable.