So long as it isn’t Celerity, yes. In other words, ONLY Celerity questing is effected by your use of a transit. You CAN use transits on ANY OTHER quest. Let me clarify still further. Getting Locate? You CAN use transits. Getting Stamina? You CAN use transits. Getting Suction? You CAN use transits. Getting Neutrality? You CAN use transits. Getting CELERITY? You CAN’T use transits. Is that clear enough?
WHY can you not use transits when Celerity? Because it cancels your quest and you lose the money you paid to attempt getting the power. What is the logic in this? The barman told you to ‘run’, for one. Also, the idea of the power, to increase the ‘speed’ of your pire, reduce the number of minutes it takes for you to gain an Action Point, logically should be considered when you have to do a quest for it.
You use far fewer APs when you use transits to cross a large portion of the city, so, as if you are training or paying a cost of sorts, you simply MUST use your own power to get you to the next place. One should note that Scrolls of Teleportation are thought of as your own power, in the game construct. That is, you ARE able to use teleport scrolls to go from one bar to the next if you wish. One player has done this on a Celerity 3 quest and used a SoTele for every trip to the needed pub. All 12. Bloody expensive.