Have we ever been told/shown how River moves about in time & space?
Whenever she 'decides' to leave the prison, she pops up wherever and 'whenever' she wants to with seemingly no problems at all.
At the end of the last broadcast episode the Doctor told her to 'make sure everybody gets home' but how and in what?
I'm sure I must have missed something and that one of you will point it out to me in kind way.

She has the vortex manipulator (I think that's what its called).

vortex manipulator

Thanks for the info- just looked it up on Wikipedia and now I know!
For a while I was wondering if she had her own Tardis somewhere.

River got her Vortex Manipulator prior to the Pandorica business. Since this is a pretty late event in her life, in her timeline AFTER the events in 1969 which was after her trip to Demon's Run (or I presume they're in this order) this leaves the question...how can she travel through time and space if she doesn't have the vortex manipulator yet? Does she have an earlier one that burns out and needs replacing?

The order of events around her appearances, and particularly at what point she went to Dorium in The Pandorica Opens, is somewhat murky.

Er......she's Rory William's daughter. The Time Vortex does what she tells it to, or she'll tell her Dad.

I'm guessing at least the part in TPO when she buys the vortex manipulator from Dorium is set before AGMGTW, given Dorium dies in AGMGTW (well at least loses his head).

Would've thought so, although Mr Saucepan has a fair point about River's personal timeline too.

Oh...and Granny, you deserve a bag of pear drops for that