Welcome to the forum!
There should be an owner's manual that came with your trailer, which will explain all the systems, including the plumbing. There are actually two systems for getting water:
1. the tank you filled, which is then connected to a pump powered by the 12-volt on-board electrical system using the batteries (or converter/charger when on shore power) -- your pump is the black object next to the wheel well at the top of your first two photos;
2. a city water hose connection on the outside of your trailer, which I think is near the roadside rear bumper.
[Edit -- see photo on PDF p. 64 of manual linked below]
There are many details to fill in, and precautions to take, but to get water from the tank to fill your plumbing system you must locate the switch(s) for turning on the pump. One switch is usually located on the general systems monitoring panel, along with readouts for your water and holding tank levels, plus battery voltage. There may also be a separate pump switch in the bathroom.
Our you can hook up your hose to the city water inlet. An inline filter and water pressure regulator are recommended. Your city water system has a pressure regulator shown in the photos (grey/blue device with brass top adjuster on the lower left of the photos) but it may or may not work correctly.
I think this is your manual FYI:
https://www.airstream.com/wp-content...ream-TT-OM.pdf
The plumbing section is "R" which starts on p. 63/86 of the PDF.
FYi there is also this recent thread about a similar 1994 34' with pump issues:
http://www.airforums.com/forums/f443...er-169136.html
Lastly -- for now -- you might like to peruse the list of topics in the Fresh Water Systems sub-forum, as there is plenty of reading available here!
http://www.airforums.com/forums/f443/
Good luck!
Peter