which would be the best route to take? That is a very good question as the answer is it depends, but here are some pointers.
The first thing to say is that all of them are good in their own way and a boat trip is a brilliant way to see London whether as a Londoner or visitor to this great city. I once blogged The Ultimate Walk by the Thames and it covers the Westminster to Tower Bridge section of the river which overlaps with many of the boat trips listed (e.g. RB1, RB2, RB6, City Cruises, Circular Cruises and Thames River Services)
If (say) you are a Londoner wanting to commute or travel between point A and B somewhere near the river then the Thames Clippers are fantastic. Compared to tubes / trains / bus, the view is better, the comfort higher and often you are less likely to need to change.
One downside to the Thames Clippers is they are often not taken into account in the route planning tools such as the CityMapper app and the TfL web site. This might be because they are often slower than the equivalent train, but for me its not that much difference in time and a lot in quality.
The best approach is to download the route timetable as a PDF to your phone and then check the times yourself.
It's also being aware that there can be problems in that there don't seem to be available back-up boats when the main Clippers break down so its worth following them on Twitter if only to be warned of these types of events:
If you have a bit longer and want a 360 degree view then one of the tourist boats such as City Cruises or Circular Cruise would be good bets. In this case go to Westminster Tube and head out to the river and there'll be an array of ticket booths and then boats to catch. City Cruises seems to be market leader but then I experienced a crush waiting for the boat so it depends which you want (note that part of the problem I had was wanting the longer Greenwich route: had I just been doing the central section I'd have been on-board much quicker).
If you just want to go down to Greenwich (or indeed Woolwich) then Thames Clippers might still be the answer even if visiting London as a tourist as there aren't that many sights downriver of Tower Bridge so you might as well get down there quickly. But if you're already on one of the tourist boats (to see the central section) then you might as well just keep going.
For the upper river the problem is the journey time: if you're in a hurry to get to Kew or Hampton Court then the river isn't the answer as it takes hours. But it is a very pleasant way to travel, so if you want to see more of the Thames, both tidal and non-tidal, to get a feel for the river's history and changing scenery from the Palace of Westminster to the palace of Henry VIII jump on board - preferably on a nice day so can sit outside without getting sunstroke or drenched in rain (again, start at Westminster tube).
Not sure about the Turks route from Richmond to Hampton Court - that seems a long way for a visitor to London to go just to get the boat, so maybe more of a locals day out thing.
If you put me on the spot and said I had to choose just the one route, which would it be? That's actually an easy question as my fav is my local RB6, starting at the centre, e.g. from Blackfriars or Embankment, passing the London Eye and Westminster and heading upriver to green and pleasant Putney.
Hope to see you on-board one day!