I'll give it a go. If your shifting was working properly before, then your derailleur limit screws should be all set. Theoretically, it should just be the case of installing the cables with the appropriate tension, but it usually doesn't work that way (or at least that's what I've found YMMV, but I tend to end up slightly adjusting the limit screws anyway)...... Also, this is most easily done using a workstand, but not impossible to do without one (just a tad more inconvenient).
Assumptions: you know and have the correct cable routing, your cables and housing are in good condition, and your cables are in the position ready to attach to your derailleurs. Your shifters are compatible with your current drivechain. When I say "chainring" that refers to the chainrings at the crank position, not the rear gears.
1. for front derailleur, make sure you front shifter is in the position where there is the least cable pull (i.e. usually where the chain is on the inner ring). Make sure if your shifter has a barrel adjuster that it is about 3 turns out (this allows for adjusting the tension of the cable). Attach the cable to the front derailleur with as much tension as possible without the front derailleur moving out of the position. You can help do this by manually pushing the derailleur towards the large chainring. -this can help you set a good tension.
Since you adjusted the barrel adjuster of your front shifter (you did, right?) this should allow for some lee way to adjust the cable to get the tension just right (this can take a little practice; it's not unusual not to have enough tension on your cable which is why I mentioned manually pushing the front derailleur towards the big chainring to help get that initial tension). Again, theoretically if your shifters worked well before, they should have their limit screws well adjusted, and you shouldn't need to touch them, just adjust that cable tension.
2. Rear derailleur: this is similar for the front derailleur, but first, put your chain in the middle chainring (this is the middle position, and should help provide a "neutral" chain position for the extremes of the inner and outer chainrings so you get the best adjustment). Again, screw out your rear shifter barrel adjuster by 3 turns and put the shifter in its shortest cable pull position. If all is working well, your chain should slip into the rear gear that requires the least amount of cable tension. You can pedal a few turns to make sure (be aware if your limit screw setting is wrong, your chain may become trapped between the frame and spokes/gears so be careful and gentle). Once that's done, this is similar to the front derailleur; attach the cable to the rear derailleur with just enough tension to prevent the rear derailleur from moving from its position. Again, you can help gain tension by manually pushing the rear derailler towards the other gears. And again, you should be able to adjust the cable tension by the rear shifter barrel adjuster.
Pretty much, so long as you have your limit screws set properly, it's then just a case of getting the right cable tension. As I said before, usually I end up having to adjust my limit screws in some little way, but you should be in the ball park with the above. And as I also mentioned before, having a workstand makes this job much easier. Finally, have a test ride -you may find you still have to do a little tweaking as compared to the no load situation of the bike being in a workstand (or at least with no rider on it trying to push down on the pedals).
Forgot to say: at least for Shimano, there is a different cable pull for the front derailleur for road vs mtb. I'm guessing you had a road group on before, so just make sure your front shifter has a road pull (sounds like it has as it is called Sora).
Hey, good luck!
