Lydian is the same as a major scale with a raised 4th step or the white keys from F to F. To play this scale on a different starting note, play a major scale and raise the 4th step. For example, F major has one flat (B flat.) Since B is the 4th step of the scale, we raise it to B natural.
Lydian is challenging. Something about that raised fourth makes it very difficult to use this scale. In fact, I started this series on the modes partly in hopes that it would encourage me to figure out Lydian a bit better. I find it interesting that even though modern music theory spends a lot of time worrying about 5ths, changing the 4th can cause even more trouble. The 4th used to be considered one of the most important intervals and even today, it is described as one of the "perfect" intervals. But it doesn't get as much attention as the 5th or even the 3rd in music theory. It is treated like a stable, reliable and immovable landmark that we can count on forever. So it is no wonder that when we change this step of the scale, it throws us for a loop. The only mode that is more trouble is Locrian (up next, may the Muses help me) which changes TWO scale steps instead of just one.
The one thing I have managed to do with Lydian is use it delicately. If you ease into it and treat the scale as a fragile mental construction, you can sneak up on that fourth without it throwing the whole piece into another key. It can be a serious challenge to avoid that fourth until the melody is strong enough to handle it but I do like the pop a successful Lydian fourth gives. It lifts the whole piece up into a different emotion.
"Robin m'aime" by Adam de la Halle (also known as Adam le Bossu or Adam d'Arras) uses a Lydian scale. This is a wonderful tune from the 1200s. It is simple and sweet with the Lydian fourth opening the melody up in gentle and unexpected ways. (By the way, "Robin m'aime" seems to have started out as an old chanson that Adam borrowed for Le jeu de Robin et Marion which is the oldest surviving secular play set to music.)