Red Orbs in Devil May Cry 5 serve as currency. You purchase abilities and items with them, e.g. resurrection stones, and Nero's Breakers.
The key to getting a lot of orbs lies in observing your surroundings - the orbs can be mostly found on small bushes that drop a given number of orbs after they are destroyed.
Pay attention to the distinctive "trees". You can find a scarce number of them throughout the in-game world, often hidden across different maps. In comparison to smaller "bushes", which you destroy with one hit, destroying such tree requires a series of the hit to drop red orbs. The difficult part is that you have only several seconds to deal as much damage as you can before it becomes corrupt. The orbs you do not collect stay on the tree and you will not be able to pick them up anymore. Thus, use your most powerful abilities and treat the tree as a strong opponent.
The main source of the currency is combat. If you perform spectacular combos, you reach a higher rank, which translates into better rewards at the end of a mission. In order to build up good combos, you should consider purchasing new skills. You can also achieve a lot with the characters' default abilities. The most important part is to use different abilities within one combo (not just one over and over again) and doing it as often as possible (do not take any brakes - jump from one enemy to another). More information about combos can be found in How to get a lot of style points? and How to fight? sections of this guide.
Initial method (V)
In my opinion, V best handles generating a high-class combo (even an SS rank) rewarded with a lot of red orbs. Moreover, you can purchase the Green / White Profit abilities allowing you to generate even more orbs from the defeated enemies when the Nightmare is active. It is also worth collecting and buying purple orbs to extend the time Nightmare is on the battlefield. After purchasing the Trigger Heart ability, the bar is reduced at a slower rate.
Latter method (Dante) - the most effective one
Upon reaching Mission 13, Dante receives a gift: Dr. Faust's Hat. It will be added to your inventory as an additional weapon with its own abilities to purchase, that increase its stats. Select a target and throw the hat at it - it will stay on the opponent's head for several seconds - during this time, attacks on this enemy generate large amounts of red orbs.