Start each level with a flower chain, especially on the larger levels aslong flower chains can be a great source of points. If the key is close to your starting position grab it, but otherwise map out a route that avoids the key (it will break your chain) and doesn't box you in while collecting as many flowers as possible. Be prepared to adapt your route as monsters appear, as some will interrupt your chain by eating flowers or creating fruit if you destroy them. Use warps from one side of the map to the other to your advantage. Scores of 40,000 (for a 200-flower chain) are well worth the effort.
Grab the Key ASAP. After completing your big chain at the start of the level, grab the key as soon as you can. It's not safe to move along rows or columns without walls without the key, or your own bullets will kill you. It also protects you from being fired upon from behind.
Collect the fruit! Fruit aren't just a good source of points (especially if collected in chains), but every 100th fruit collected also awards an extra life.
Grab Red potions whenever possible. In particular, grab at least one red potion as soon as you can each life: it doubles your firepower. Additional red potions help too, by increasing your rate of fire.
Grab Green potions when your health is low. It seems obvious, but when you're low on health a green potion is almost as good as an extra life.
Always be grinding. Try staying charged up as often as you can by grinding against the walls. (Grinding is also a good way to ensure you don't miss turns.) When you start flashing red let yourself cool down so you don't explode, but then get back to grinding again so you can squish enemies.
Squish Puddings for massive points. Puddings can be your greatest source of points in the game, thanks to squish multipliers. Get charged up and find a nice long wall with a nearby pudding. Bear down on the pudding while charged: your bullets will cause the puddings to multiply, setting yourself up for some massive squish bonuses as you barrel into them -- not to mention all the fruit and potions they drop.
Squish Centipedes for fruit by getting charged up and running into the head of the centipede. When you squish the head, ALL the segments of the centipede turn into delicious fruit, setting you up for a valuable fruit chain and progressing you to an extra life.
Create Irises while collecting flowers. Activating a Bomb or Bob-Omb while working on a long flower chain can create several high-scoring Irises. It's too tricky to pull off as a reliable high-scoring strategy, but it's immensely satisfying when you can do it.
Don't complete levels too fast. Instead, take the time between when you collect the last flower and when you exit the maze to destroy enemies and collect their tasty, tasty fruit. You may even want to leave the last flower uncollected, to avoid accidentally exiting the level by bumping into the Door. When the Ghost appears, make your exit (or if you're good, keep on farming enemies for fruit while avoiding the Ghost).