The sword you begin the game with, the Rebellion, is a quick, yet powerful weapon that'll suit you well throughout the game. Not many enemies are going to be specifically weak against its plain magic enchantment, but it's nice and quick and can hit multiple enemies with its default combos, provided they're grouped together.
Combo I: T, T, T. Simple three-swing maneuver that will usually knock weak enemies down. Continue to tap away for extra damage while they're on the ground. Excellent when used in the middle of a crowd of enemies; the wide swings will normally hit multiple foes.
Combo II: T, pause, T, T. Another forward-aiming combo with an extra swing.
High Time: Hold R1 to target, press back on the left analog stick, then hit T to launch your target up into the air. If you hold T, Dante will jump up after the target, allowing him to swing away in mid-air. Either variation can be a good starting point for a gun-assault; Ebony & Ivory can often juggle an enemy in mid-air indefinitely. Not all enemies will be launched upwards by this maneuver, though.
Helm Breaker: Hit T while in midair for a downward slash. This can be performed either after jumping or after holding down T to launch yourself during a High Time.
Stinger (2,500 Orbs): Hold R1, press forward on the LAS, and hit T. Dante will launch himself forward towards his enemy with his sword outstretched; if he hits the enemy (he might not, if the enemy moves or if it's too far away), it'll usually be knocked backwards and onto the ground.
Stinger: Level 2 (10,000 Orbs): This is an upgraded version of Stinger. When performed, Dante will hit with more power, and will be able to travel across a greater distance.
Drive (10,000 Orbs): If you hold down T, Rebellion will begin to glow; releasing it will send a shockwave towards a nearby enemy.
Air Hike (20,000 Orbs): Although expensive, Air Hike is going to be something you should save up for early on in the game. After you jump, Air Hike will allow you to double jump by tappping X again while you're in midair; this will effectively double the maximum height of your jumps, allowing you to escape from tight spots and stay off the ground for a longer period of time.
Million Stab: While performing either a Stinger or at the end of a Combo II, start rapidly tapping T, and Dante will begin repeatedly stabbing forward with his sword. Although this move doesn't have much of an arc of damage, it will do massive damage to anyone in front of Dante, and if you keep hitting the button for a while, he'll eventually finish it off with a single powerful thrust.
Cerberus is a set of cold-enchanted flails that Dante picks up early in his adventure. Although they have a slower attack speed than Rebellion, they do a decent amount of damage and can be used to strike enemies on multiple sides of Dante in short order.
Combo I: T, T, T, T, T. This simple combo will knock around enemies on all sides of Dante.
Combo II: T, T, pause, T, T. Getting the timing right on this can be difficult; you need to start the second set of taps just as Dante is about to place the weapon on his back. If you can pull it off, though, he'll conclude the combo with a Windmill attack. This isn't necessarily any better than Combo I, save for the fact that you'll need to do it to pull off a Satellite.
Swing: If you jump and hit T, Dante will swing at any nearby aerial enemies. Unlike the Rebellion's airborne attack, this won't hit any enemies on the ground, making it less useful.
Revolver: Hold R1, press forward towards your target, and hit T. Dante will swing around in midair, causing damage to all enemies in front of and behind him, then wind up hitting the ground with Cerberus. Even while locked on, this has a good chance of kind of warping around a stationary enemy, and doesn't revolve all that quickly.
Revolver Level II (15,000 Orbs): A much improved version of Revolver, level II will let Dante make more revolutions and hit for more power.
Windmill: Hold R1, hold back on the LAS, press T. Unlike most weapons, this sequence of presses for Cerberus doesn't launch enemies up into the air; instead, it'll send your weapon forward in a whirlwind maneuver, lightly damaging enemies immediately in front of you. Not a tremendously impressive move, and it's a bit slow for the damage.
Satellite: If you can pull off a Combo II, then start madly tapping away at T after the pause and you should wind up sending your flail rotating around your body, damaging enemies on all sides. Unfortunately, the damage radius for this is fairly small, so anyone nearby will have to basically be right on top of you before they get hit.