Select the planet upon which you want more slots.

On the lower right is a button that looks like a planet.  Press it.

A new list of options will appear.  One looks like a grear.  Mouse over it and it will read something like "Logistics"

Press this(Assuming you have enough credits/metal/crystal) and it will increase.  Most planets have multiple levels of this, each costing more than the last.  I believe the Terran planet has the highest number of levels, 3, maxing out at 24 logistics slots.  Asteroids max out at 12.

You can upgrade Tactical slots in the same way.  The button is just to the right of the logistics one.  Read the other upgrades there as well.  The first upgrade you get on a planet should always be Civilian Infrastructure, so it doesn't cost you money and gets you income.  Also, don't forget to build extracors as soon as you colonize.

-Twilight Storm

His title is logistics slots, but yeah, good to clarify if he's confused.

Yup, that's exactly it.  Although Ryat was correct.  (Dangit )  There is a TEC research that gives you +4 Logistics slots on every single object you colnize(Which is esspecially useful for Pirate bases and dead Asteroids, which I believe do not get Logistics slots at all without it.)  You'll find it in the Civilian Research tree.  Similarly, Advent have a similar research, except for Tactical Slots(+5 I believe)

-Twilight Storm