Every type of ship costs some logistics points, like 2 per tiny ship, and 10 for huge ships (the biggest ones). Techs and buildings increases the cap to maximum 60-80'ish meaning if you are at 60 logistics, you can put 30 tiny ships into a fleet, or 6 huge ones, or a mix of everything until you spent 60 logistics points.
As for what to research, the tooltip should show you all techs which gives logistics, and they are found in the second tree from the top (called engineering if you play terrans). They are generally called something with logistics, and shouldn't be too hard to find
There is one more advanced trick to getting more logistic points, and that is the building 'Hyperion Logistics System'. You can only have one of these, so you should plan how to use it properly. Meaning you want to surround it with military buildings like the Elerium Defense Shield (+3 to military buildings), and anything else that gives bonuses to military improvements. The higher you can boost this building, the more logistics points you can get. It gives 1 more per level, so if you can get it to level 10, you would get 10 more logistics points etc. I think the highest I've managed is 17, but anything around 10 is fine really.
(the Hyperion Shrinker building works the same way for putting more guns on your ships)