Pregnancy test will tell you whether you are or not pregnant as early as 3-4 weeks after you've miss your (if regular) period.
By the time you begin to gain weight, you will probably be at least 12 weeks in gestation. Again it depends on individuals. Some people hardly gain any weight, all the weight comes from the fetus, increased blood volume etc. And after delivery, within 6 weeks, they are back to their pre-pregnant weight.
What do you mean by things moving around in there. Nothing moves around in there!
Its in your uterus that the fetus grows. Everything is anchored very nicely in your abdomen, all our internal organs are. The same goes for your uterus. It does not move around.
At 5 months of gestation, you may feel a bit of flutter in your abdominal area, that's your baby moving in there. Second time (experienced) mothers feel it earlier, at about 4 months.
Hope this helps.