If you want something that is relatively intuitive, you can't go far wrong taking some inspiration from the MMORPG or other level based games (e.g. World of Warcraft) which give each character an experience bar to denote their level and indicate how far until the next one.
The basic layout is something like
(BRONZE) ===============|--------------------------- (SILVER)
CURRENT (35/100) 75 Airmiles until next level. **shiny!**
This basically depicts that the current level is on the left, with the desired level on the right. With a graphical bar representation of the progress stretching between (denoting the percent complete). Then either within, on-hover or beneath, the fine grained detail about how many air miles they need to complete until the next level.
While not probably relevant for flying, if you have consistent increments, you may find it better to depict as those instead. E.g. flights probably differ in the number of airmiles gained per flight. But if you do it simply on quantity of flights (rather than distance), you could say "6 more flights until next level". If you so wish, you could change the bar into an appropriate number of icons (e.g. aeroplanes) that "light up" (un-fade) as you progress through the level.