Xenon is mostly used to make blue lamps.

The following paragraphs about the use of Xenon come directly from Nature's Building Blocks: An A-Z Guide to the Elements by John Emsley:

"When an electric discharge is passed through the gas is produces agentle blue light, which extends in to the 'safe' ultraviolet region as well, and so can be used for sunbeds and the biocidal lamps seen in food preparation areas. Xenon 'blue' headlights and fog lights are used on some vehicles and are said to be less tiring on the eyes. They also illuminate road signs and markings better than conventional lights.

Other types of lamps use xenon's ability to deliver and intense burst of light when pulsed with a very high voltage, and so it is part of the built-in flash of modern cameras, as well as being essential for the strobe lights used in high-speed photography, to record such events as a bullet emerging from the barrel of a gun or striking a target. Xenon flash lamps are also used to activate ruby lasers."

"Although it is almost totally inactive chemically, xenon is used for space flights because it makes the best fuel for ion engines. In such an engine, a beam of ions is accelerated by and electric or electromagnetic field and then expelled from the vehicle at 30kilometers per second (about 100000 kilometers per hour), thereby giving the vehicle a powerful thrust in the opposite direction. ... Anion engine is 10 times more efficient than a conventional propulsion unit. So-called xenon ion propulsion systems (XIPS) are now used by at least 10 orbiting satellites to keep them in the correct orbit and each has enough xenon fuel on board to last more than 12 years. The space probe Deep Space I is powered by a xenon ion engine."

It is hard to say exactly how long it too Ramsay and Travers to find Xenon, because it is difficult to define exactly when they started looking for it since they did not actually know it existed until they found it. In 1894, Ramsay helped to isolate argon, and then in 1898Ramsay and Travers discovered neon and krypton. Later in that same year they also discovered xenon. Hopefully that gives you a sense for their research timeline.