I understand it is a medal system but I'm really new to shmup styles of games. How exactly does the medal system affect the multiplier?
Each medal is worth one more than the last one. The first one will increase your multiplier by 1, the next one by 2, then 3, 4, etc. Thus, collecting n medals will give you a multiplier of n*(n+1)/2. E.g. after 100 medals, you'll have a multiplier of 5050.
Some enemies will give you medals worth a little more (+5, +10 and even +200). For example, collecting a +10 after 2 normal medal will give you a multiplier of 1 + 2 + 3 + 10 = 16.
There are a few points to be noted : losing a life will return your multiplier to 0, collection a power up (the big "P") while at maximum power will increase the multiplier by 100.
Here is a more general guide on Giga Wing : http://shmups.system11.org/viewtopic.php?p=73570
Here are two videos of very good players that will show you the good spots to reflect : http://super-play.co.uk/index.php?content=2&gameid=99
(The players in these videos are very good, you won't be able to mimic those paths on your first few tries :) )