You definately need to research a specific technology to get Combat Transports, which BTW are the nicest looking ships you can build IMHO. There's a thread running around regarding a two-page tech tree you can download that should tell you specifically what tech to research to gain this ability.
Std TR's are available through Propulsion Theory --> Cold Fusion --> Impulse Drive.
Combat TR's take BB Technology: Propulsion Theory -->Artificial Gravity; Defense Theory --> Deflectors --> Shields; and Weapons Theory --> Phasers --> Energy Channeling --> Interstellar Tactics. These are all pre-requisites to Battleships, and will allow you to research that tech when you have completed them.
BB's are a very high-cost tech (2500 beakers), but they are a "new-generation" starship which render practically every previous design obsolete.
Actually, unless I'm planning on using them soon, I usually stick with standard transports, since the maintenance on them is 0 compared to 5 for the combat transports. However, if I know I'll be using them soon, the 5 standard transports it takes to match the capacity of one combat transport takes 2.5 times the production.
I used to do that too. Then I realized that the CTs make pretty dang good defense ships. It takes a pretty determined enemy attack to wack a CT. Since you can't park them in orbit without unloading I typically park them on my crucial starbases, especially near the frontier. They serve the dual function of giving me solid defense (8 CTs can pretty much render a starbase invulnerable until Overlord times), and a nice counterpunch for invading offensive enemy worlds.
And not only that, AkTR's can be "parked" outside a poor PQ planet to "liftoff" malcontents without wasting IU's on an (otherwise) useless Colony ship. They've got enough MF (and DF!) to simply tool around new colonies filling up. Once at 5k, they're roaring to go (and your morale is at +80!).