They don't like you to apply much rpm to pull away, the ecu does it for you.
It's something a lot of owners have complained about, a lot of modern cars have an anti stall system when pulling away, so they detect the movement of the clutch and ups the rpm to save it stalling, it also helps smooth out gear changes as it will of course, rise the rpm a little everytime you dip and release the clutch.
You can hear/watch it do this just by sitting there with the brake off operating the clutch pedal at idle.
Fiat seem to have used a simple (cheap) system and it seems it has caused a few headaches, mainly with (older) drivers that are not used to it.
(my lad can't pull away in an older car that doesn't do this!)
As the ecu ups the rpm and you are also applying some throttle and it bogs down a little because of this, it doesn't seem to realise you are inputting the pedal (and return fueling control back to you) until you are down the road a little and it's finished "helping".
Try allowing the ecu to get you just off the line before applying the throttle pedal, lift the clutch and feel the rpms rise, as it pulls off then apply the gas.
It becomes quite easy to master after a while and it'll even do it on very steep hills without problem.
As for your other issue, it does rather sound it's being flogged and stuttering in too high a gear, particularly as you say, changing down or speeding up stops it.
You might be a little too easy on the throttle trying to "run in" the engine.
