I find it a bit hard to approach aggressively as the Pits. It's certainly possible to play a very fast paced game, but it's not really a "in-your-face" kind of fast play. Dash attack and b-air are the only moves that make use of your good disjoint while being fast at the same time, but DA is unsafe and b-air is a good kill move so you might not want to stale it. Like knepsis said, grab follow-ups are really good, but that's more like a punish option than an approach I think. Arrow approach is a bit too slow.
When I approach with Pit, I generally try to poke with f-air a lot by default. There's a bit of startup, but once it's out there aren't a lot of characters that can beat it unless you go in too close. You can do many things, like short hop forward then retreat f-air, neutral jump (then retreat if needed) f-air, or even jump back then advancing f-air. The last one is a pretty good mix-up, especially if your opponent wants to close distance, but don't do it too often since advancing f-air is never safe. If your opponent is starting to challenge your f-air with their fast aerials, try jumping in deeper and using n-air instead if they look like they wanna do it again.
If your f-air hits a shield, you generally have time to land before the enemy can punish, unless they have a tether grab or something (mainly Lucas, since his tether is fast). I don't know the specific frame data, but from my experience they can act out of shield before you can do anything; but, if you spaced properly, the distance between you two means your disjoint can still be an advantage in some matchups. I tend to use jab or f-tilt right after landing, but sometimes a defensive option like shielding or just jumping away might be better. You can even do jump back advancing/retreating f-air I guess.
Keep in mind a lot of these things might not work in some matchups...like ROB. But the general rule of 1) f-air 2) grab still stands.