I think the Park Tool video series on derailleur adjustment is pretty good. Here's the one covering front derailleurs:


The only thing I'd add to that, especially since you're concerned about cable tension, is after you get the L limit screw set, that's generally when you would adjust the cable so that there's no extra slack when the derailleur is sitting against the L limit screw and the shifter is in the 1 position. If your cable has slack here, then you're probably not going to get good shifting up to the middle ring. I think Park Tools' rear derailleur video advises undoing the cable adjustment bolt entirely when you set the limit screws, and I think that's a good idea for the front also. Undoing that bolt ensures cable tension is not influencing your limit screw adjustment, and it brings it back to a baseline situation like you're in -- where you've fitted a new derailleur to the bike. So, I would:

1. Adjust the physical derailleur installation like they show in the video.
2. Set the L limit screw to about where you think looks right.
3. Attach the cable and pull somewhat taught when you affix it so you can do the L limit screw testing, like they show in the video.
4. Once you're happy with the L limit screw adjustment, loosen the cable bolt and re-tighten, pulling all slack from the cable. Fix the cable bolt and test the L limit screw again per the video instructions.
5. Move to the U limit screw like they show in the video.
6. Move to index adjustment like they show in the video.

At that point, your derailleur should be aligned correctly on the bike, the L and U limit screws should be set correctly, your cable tension should be set correctly, and your barrel adjuster on your shifter should be adjusted correctly to get relatively good indexed shifting. Front shifting is usually not precision art; your goal is generally to get relatively consistent and reliable operation and it can be done with some practice or trial-and-error. If you can't get the indexing correct and just generally don't like it, you do have some options for non-indexed front shifters, which many people prefer in general (due to the more finnicky nature of front shifting). You can get these in twist shifter configurations and in thumb shifter configurations. With a non-indexed shifter, you don't have the 1/2/3 positions -- it's just a bunch of smaller clicks during the range of motion that allow you to stop the cable movement anywhere along that spectrum, to correspond where you want the front derailleur to be. In practice, this usually works pretty well for front shifting.