I think the two most common combo starters for Fox are shine and grab (up-tilt is nice but it's more situational, even more so for down-tilt). In general, you want to keep your opponent in the air to juggle them so you'll want to usually go for things like up-throw to up-air or SH nair chains.
For shine:
If your opponent's character doesn't get knocked down by shine, you can wavedash in their direction and either:
1. Dash->grab
2. Dash->SH nair
3. Dash->up-smash
4. Dash->SH dair
5. Waveshine
4 is very SDI dependent, while 1 and 2 are more consistent for following up against SDI. 3 is usually used as a finisher, but can also be used to send the opponent into the air to juggle them but depends on their DI. 5 only works against characters like Falcon, Peach, Link, Ganon and is also very SDI dependent.
If your opponent's character does get knocked down, you can wavedash in their direction and either:
1. Jab reset if they don't tech
2. SH up-air to cover tech in place or no tech
3. Dash to tech chase and have an opportunity to grab
For grab:
The basic followups for up-throw are:
1. Regrab (if they're spacies)
2. Up-smash (if they're a fastfaller)
3. Up-air
4. Weak hit full-hop bair
5. Chase with bair/nair as a finisher
6. Weak hit reverse full-hop/DJ nair