The target painter doesn't have any effect on the target range or the number of targets, but it does make the signature of the target ship larger which makes it much easier for other ships to target it. I don't recommend getting this until you're in a more advanced stage of fleet battles.
To help out your fleet's targeting, grab a Remote Sensor Booster like the Linked I Sensor Network. Target your fleetmate and activate the booster on them to give them a bonus to max targeting range.
That should take less than 3 days to train that up. The electronics skill will obviously take the longest time, and you'll want to continue training up the rest of those skills to increase your bonuses.