Every time you kill an enemy in Zero 1 and Zero 2, you gain EXP for the weapon you used to attack. Depending on the weapon, you might have alternative means of attacking. Each of these alternative attacks have their own EXP pool to level up and unlock new combos. For example: For the Z Saber there's a separate EXP pool to unlock the Rolling Slash as there is to unlock the Triple Slash.

You can monitor your progress in the "Sub Menu", such as in this screenshot here:

The stars next to the weapon name represent your mastery level. Each time you level up the weapon, you fill in one of these stars. All mastery stars are visible from the start, so you can tell how much progress you have by how many of these stars are filled in and how many are still blank. If any stars are still blank, that means you still have work to do in some regard to finish mastering the weapon. If you have different types of attacks you can use with the same weapon, grind using each one individually to unlock it's extension.

In the early section of the game after you first arrive at the Resistance Base, head down to the Transerver and return to the Underground Lab. This is the area you began the game in. Backtrack through until you reach this spot:



From here, you can begin grinding your saber and buster. The advancements you can make at this point are as follows:

Saber:
Single Slash > Double Slash > Triple Slash > Charged Slash > Faster Charge Speed
Jumping Slash > Spinning Slash
Dashing Slash > Rolling Slash

Buster:
Three Bullets > Four Bullets > Charged Shot > Faster Charge Speed

I recommend killing as many enemies in one hit as you can to make this go by faster. You can off screen enemies a bit to make them respawn, simply run back and forth a bit to make the spider nest respawn. You should be able to finish grinding this up in about 20 minutes.

Later, when you have done the mission to return to this area and it gets blocked off, you can do grinding on the surface instead, in this area off to the right:



This is a great spot to grind up your Triple Rod and Shield Boomerang once you have them. There's two tower enemies here, one to the left and one to the right. Pick one and keep killing the extensions that automatically respawn. I recommend trying to get your shield to orbit you for maximum efficiency, but if you can't pull this off then just keep throwing it.

Unfortunately, I don't know about any places you can go in Zero 2 to grind up levels quickly, you'll have to use what you've learned and find a spot you are comfortable with. I recommend not doing this during missions, as the time it takes to grind up your weapons could negatively impact your rank at the end. Instead, warp back to a location after you've finished the mission there and do the grinding then. Apply the same knowledge you learned about Zero 1 and you should be able to max everything out in Zero 2 as well without too much trouble.