As a passenger aboard a taxiing aircraft, sometimes I feel like due to the airport layout, we're taxiing for ages (especially after landing when I want to just reach the gate ASAP already).
It's very rare that you actually have to taxi (after landing) for a huge amount of time. The biggest delay happens when you have to wait for something, especially an occupied gate. Landing on 24R in LA, it took just on 5 minutes from touchdown, to reaching the gate.
What airports are best and worst at minimising the taxiing distance between gate and start of takeoff roll/touchdown (at least for the most commonly used runways)?
Distance is a consequence of their layout, and I can only think of one where it's more than a few minutes drive from the runway to the terminal, and that's 18/36 in Amsterdam. 36L in particular is a fair way. On the other hand, aircraft taxi at up to 60 kph, so even that 8 km taxi should only take 10 minutes.
Waiting to cross runways can be a huge delay, though generally when the wait is long it also means that there's major congestion around the terminal. I've seen aircraft waiting to cross for over an hour at JFK.
As pilots though, we sometimes intentionally add to the taxi time. Eek, you say. Well the reason is that the engines have a minimum time at idle before we can shut them down, so you can sometimes juggle the taxi speed to have the cool down delay expire just as you reach the gate. Otherwise, we'll have to sit at the gate, with engines running, and you all know how good people are at staying seated once at the gate, even if the seat belt sign is still on.
Which airports are otherwise best and worst at being able to taxi around easily and safely - eg. good or bad signage/lighting, crossing busy runways or taxiways, errant vehicles, any other structures on the ground?
Most places are pretty standard. The least standard was Vancouver, but it was also the best, using neon signs as marker boards. Very clear. The worst is probably Tokyo Narita, which is just like taxiing in a sea of blue and green lights.