The easiest method of spamming is to assign your most spammable spell/ability to down+button, then simply holding down and mashing the button and backdash. The speed that this can make attacks come out is sometimes faster than the speed that the game registers hits, so it is a good idea to pace yourself to more efficiently use MP. It is possible to spam nearly as fast with the other directions if you have the dexterity and rhythm. Generally, cancelling normal attacks is a lot trickier, but not particularly necessary.
The base strength of a character is generally determined by the level of their skills in total, so the way characters powerup differs. Most characters, Belmonts, Shanoa, Yoko, and Maria have to grind their skills/spells up through use (more on this later). Soma needs to absorb and level up souls. Alucard needs to collect multiples of his spells. Charlotte needs to level up her spells by absorbing them with her shield. Doing this not only makes the specific skills stronger, but raises the character's base strength. This doesn't affect the look of any of their weapons except Charlotte. A Belmont's whip will always be their whip, but with proper grinding they can do the highest single hit damage in the game with just their whip and martial arts. Soma and Alucard are very gear dependent for their damage, so it is less important for them.
Shanoa is a whole different thing altogether. She needs to absorb her spells from the witches (Aliosomethings) and nova skeletons, and I guess she has some newer ones she can get from chests in later chapters, but getting them multiple times doesn't empower them, grinding their use does. She has a few different normal attacks that generally are gained from boss chests. I'm partial to pnuema because it is reppukens, and properly timed use of it in both hands yields double reppukens. Also it is good DPS if backdash and crouch cancelled carefully and rhythmically.
The best way to grind skills is to get a friend and play the Survival Hard Mode. You will be locked in a room with Gergoth and the other guy for three minutes. Use the previously explained spamming method to throw some shit at Gergoth until you are out of MP. Since you respawn immediately in this mode you can just kill yourself to regain your MP and spam all over again. In most cases the Gergoth will die before the time is up, so there is some downtime, and it will take a bunch of runs to max out a single skill, but this is still the fastest way to grind by far.