I have several approaches.
My bog standard approach would be a running shffl'd nair. Obviously its good to throw some tricks in there like wavedashing, dash dancing, short hop waveland etc to keep things varied.
My next approach would be a running JC'd grab which often leads into up throw to uair, or down throw to up smash or tech chase.
I then have some lesser used approaches like shffl'd bair or dair. I use bair sometimes because it still has high priority but i generally use it when my opponent is running in the opposite direction so i hit them when their shffl misses me. dair is usually into a shine for combos or if i want to kill, then into upsmash.
One that i use quite often is my basic approach into a grab. So I can run to shffl'd nair into grab. At low percentages you get a grab provided your first hit connects. If they shield then the grab is quite quick to nab them out of their shield. If I notice they shield grab well then I use shffl'd nair into shine into JC'd grab. The beauty about this is that they get shined if they try to shield grab and you grab out of your shine for a very quick attack leaving very little options for your opponent.
A good idea is to watch a lot of different fox vids to find approaches and try to copy them. This will give you a good basic understanding of what you can do against people and then you should try to incorporate your own style into it.
Here is what I would do for the certain characters you mentioned:
Fox: I would grab more often than usual because you can chain throw fox with fox's up throw. Up throw to upsmash/uair is great too.
Peach: I would try to approach as little as possible because peach is very dangerous to fox. I would use alot of lasers, play a controlling game. When I get an opportunity to attack, I would shffl'd dair to waveshine, to upsmash. Peach has high traction so your waveshine should leave you right next to her, perfect for upsmash. If you're really good you can tack on an uair after the upsmash too.
Falco: Falco has a huge advantage here with his lasers, so I would go as offensive as possible. stay as close as possible so he doesn't have room for lasers. As with fox, lots of chain throws and up smashes. uair isn't quite as effective in general as falco's dair has huge priority. only use uair if you definitely know its going to hit e.g after a high % upthrow.
Marth: For marth you must play control aggro. That is when you're far from marth, use as many lasers as possible since marth has no long range attacks. As soon as his within range for his sword, either get away asap or go very offensive. You dont want to be tippered. Beware of going offensive though as marth can chain throw fox very well (provided the player is decent)
For your last questions: generally you should wavedash out of your shine (waveshine) everytime unless you want to do something special like grabbing out of it or upsmashing out of it. This is usually when someone shields your shine. Drillshining should be used when you want to start up a combo, but dont rely on the drill as it has medium-low priority. Nair is better at lower % and Dair at mid %, obviously at high % you want to kill so dair to upsmash or dair waveshine (drill waveshine) to upsmash is quite good.
hope it helps
