Which abilities are worth to be purchased first for Nero in DMC5?

Nero has many abilities available for purchase - especially those improving his sword fighting. Firstly, consider purchasing cheaper abilities: Speed and Air Hike. Both improve your exploration capabilities, which is important in Devil May Cry 5 to locate secrets, red orbs, and secret missions

Other characters can by default perform the double jump - you can perform it as Nero by bouncing off the walls.

As far as combat abilities are concerned, first focus on the Red Queen abilities. Purchase abilities slowly, one or two at a time, to master them well before you add new combos to your arsenal. The abilities you should purchase first are:

A strong attack that cuts enemies in the air and on the ground.

1. Split - Nero's basic ability, which allows you to finish an attack in the air by cutting down. Hurts enemies in the air and on the ground.

Streak also works in the air and allows you to reach an enemy on the ground to prolong combos and deal damage.

1. Streak - allows you to jump towards the enemy; useful in combat, as it prolongs combos.
2. Roulette Spin - a simple air combo that additionally drops the enemy to the ground making him vulnerable to the next attacks.
3. Red Queen Combo - there are many combinations available for you to equip, however, try not performing a single one over and over again (firstly, it's boring, secondly, it does not result in generating more style points). Purchase combos one after another, after you master them - if you purchase several at once, you will end up not using some of them in the end.

It is also worth to focus on the Swift Dodge and Wire Snatch abilities from the start (the latter extends the wire range, which proves extremely helpful when fighting flying opponents and mounting interesting combos).