The multiplier comes from killing an enemy with shot while the enemy is in contact with powershot explosions, +1 multiplier per explosion. It also multiplies the number of items that appear. When the item counter in the top corner is maximum (flashing) you can chain the multiplier between enemies without having to powershot them individually; the chain timer is short, but you can sustain it by picking up lots of items. (Technically you can always chain, it's just not very useful without max items because it'll only multiply the base enemy value and not affect the items)
So powershot enemy -> immediately finish off enemy with normal shot -> grab items -> if you have maximum items try to quickly kill more things for the chain, otherwise focus on setting up for the next.
Mostly this comes down to knowing how much health enemies have. One trick to make it easier is to memorize when enemies are going to enter the screen and full powershot them when they're right at the top. The explosions will stick, but the enemy will be invulnerable until it moves down a bit, so if you time it right this is a guaranteed x16 even on weak ones.
There are other things to the game, but surviving and getting used to the basic combo should be the main concern. A no miss run will get at least 9 million from the game clear bonus alone.
Edit:
Hazuki wrote:
I remember there being a PC shmup (probably doujin) that was apparently inspired after Darius, but the bosses were modeled after some kind of exotic sea life. I don't remember what specifically, but think of stuff like trilobites/isopods, maybe.
Pretty sure someone created a thread about it here years ago, but I can't find it anymore. And using Search for "Darius" returns over 500 results. Quite hard to find it again like that.
Cambria Sword? It's an interesting experience, albeit really really long.