Do I need to solder wires to my tilt sensor?
I don't have the starter kit but I have a tilt ball sensor:



My guess is that there is a ball inside it, and it connects the wires, like this:



This is confirmed by measurement.



I suggest you either solder wires to it, or push the wires into your breadboard. As you can imagine from the diagram, the contacts will be closed if the ball (or mercury or whatever it is) is downwards. In other words, if the leads are facing down. As you tilt it, the ball will move away from the two wires, breaking the connection. You may have to tilt it quite a lot for this to happen.



(Edited to add)

I see from the Arduino Starter Kit page that the tilt sensor works like this:



This shows that my guess was pretty accurate. :)

There are four pins, so I imagine that the ball would connect four of them if held upright, and various combinations if tilted forwards or sideways.




  If you insert the pins into the board the sensor will always be standing and there will be no way to turn it.


Well, you could tilt the breadboard, yes?

However I agree that if you want to tilt just the sensor, then soldering wires to it would seem quite sensible.

I think the starter kit is there to show you the concepts. There is nothing stopping you tilting the entire breadboard to see how it works. In practice you would mount the sensor in such a way that it activated when moved from position A to position B, whatever that is.