Although the coolant temperature may be up to normal in about 5 minutes, it takes a lot longer for everything to get up to temperature, especially the oil. It is more like 20 to 25 minutes.
You don't need to allow time for warm up. Once the idle is reasonably stable you're just wasting gas to sit there going nowhere. Roughly 30 seconds should be more than adequate most of the time, even in sub zero weather. Do drive it calmly at least until the coolant temp comes up to normal.
You didn't say what kind of transmission that doesn't work well until it is warmed up. If it is manual trans there are two things you can do to improve cold shifting without doing a trans overhaul to upgrade the shifter fork and other internal parts. First, you can use either a synthetic trans fluid that doesn't thicken as much when cold, or you can use a trans fluid with a higher level of friction modifier or both. The Ford honey with 2 oz of friction modifier added works very well as does Chrysler spec ATF+3 (alghough not synthetic, it has a higher level of friction modifier to start with). The second thing you can do is get some oil slinging around inside the trans before you try to shift. Do this by starting the engine, moveing the shifter to neutral, let out the clutch, and blip the throttle two or three times to about 2000 or 2500 rpm. After you do that and have some oil on the shift forks it will be easier to slide into gear.
If you have an automatic trans that doesn't shift well cold you probably have some trans problems. It may help to change the trans fluid, but you might have more wrong than a fluid change can cure. It is easy to do a fluid change on our cars and is worth trying before spending money for deeper problems.