Like always in Aurora, there are multiple ways to do Sorium harvesting from gas giants depending on your situation.
Firstly, it is possible to make a "gas station" style Fuel Harvesting Station that is literally just a refuelling HUB, lot of fuel tanks and a bunch of Sorium harvesting modules. Once towed to a gas giant, it is left there for good. Now all your ships can use this to top up their tanks as they move through the system or when their conditional order (refuel at a hub/refuel at a colony or hub) kicks in. This requires the least amount of micromanagement and I would recommend doing this for every transit system that has a suitable gas giant as your empire expands if at all feasible. Just remember to tick it as a Tanker in the design screen!
Secondly, a cheaper and more low-tech solution is to build a harvesting ship. It does not need to have a lot of fuel tank capacity because it is moving under its own power. A couple of cheap commercial engines, a couple of harvesting modules - the amount of course depends on your shipyard size - and you can send them to a gas giant that has Sorium. Just make sure they have the conditional order for "unload at nearest colony when full" and set their reserve fuel level at something like 2% or so. You can then leave them on their own. The downside of this system is that you are paying resources and wealth for engines that aren't strictly necessary and you do still need tankers to haul that fuel elsewhere because the ships are not smart about where to unload their fuel.
Thirdly, you can build a fleet of Fuel Harvesting Stations that are towed in place but instead of a refuelling hub, they only have the refuelling system. This is a lot cheaper module than the hub. Combined with the 'tanker' tick box, it allows each FHS to fuel up a single ship at a time. Use tugs to tow them in place and then build a cheap-ass, slow tanker that eternally goes between the FHS-fleet and your nearest important colony or even Earth. It requires some fiddling and math but it is possible to reach an equilibrium where you never get error messages about the FHS getting full or the tanker being unable to fill up. This is a great way to get fuel if a rich gas giant is in an awkward location, someplace where you don't want to send your battle fleets to refuel.
Fourthly, you can and probably should do a combination of all three - building "gas stations" in important transit systems so your ships can top-up as they proceed through your Empire, have a couple of harvesting ships cleaning out gas giants that only have small quantities of Sorium and are out of the way, and finally have a major FHS-fleet that collects the majority of fuel that you need, which is then transported by a separate tanker fleet(s) to your shipbuilding colonies and major Fleet bases. Naturally, in early game you might only need a single harvesting ship but fuel is one of the famous three crunches that players usually experience so it is best to plan this out and execute that plan well before your fuel reserves are depleted.
