Do what Goa said, it's on the research menu, below Warfare, Fortification, Empire and Manipulation there should be something called Fleet Logistics. From here you can increase your supply (bottom), which will increase your supply but also increase your upkeep, and cannot be undone, so make sure you're ready to do it. The top one increases your command, allowing you to build more capital ships / a titan. It has no effect on upkeep, but has a high one-off cost.
There's also Phasic Transmissions on tier 6 of the Empire tree. This will increase your supply by 8% at stage 1, at the cost of 1400 credits, 250 metal and 400 crystal, and 15% at stage 2, at the cost of 1500 credits, 275 metal and 425 crystal. This will have no effect on your upkeep, so it's great for the late game where you don't want to have more upkeep but need / want more fleet. This is ony avalible for the Vasari, and so gives them an advantage in the medium to late game. It's probably one of their best techs.
Finally, there's the Supply Pact, which is the final pact for the TEC. Diplomacy isn't a big part of Sins, so it rarely gets used, but if you have a TEC teammate willing to do it, the supply pact will give both participants +100 supply and -15% upkeep, which is huge in the late game when upkeep levels can reach over 80% - the Supply Pact can almost double your income, as well as letting you build more stuff.
And that's it. Primarily you will be using the Fleet Logistics research, as it's the easiest to get. Don't upgrade it too much or too early though, as you will find your income rate is decreased and you can't afford to use the supply you got - it's worryingly easy to do.
