Can you move for one action to a city second action pick up a water or train token then immediately use the water or train token?

Nope. The "Prepare for travel" action allows you to get one ticket (Ship or Train) if certain conditions are met (see Reference Guide, pag 9, Prepare for Travel Action subsection).

You're allowed to spend a ticket only immediately after performing a "Travel" action (the timing on how and when to use tickets is detailed in the Reference Guide, pag 12, under the "Travel Action" subsection)