2 prong to 3 prong outlet - with pigtailed grounds
You've got a pretty good idea on what needs to be done. Naturally, you'll be turning off the breaker to the circuit.
In cases like this, I've always pigtailed the blacks together and had one connection to the outlet and do the same to the neutrals. You'll need some extra black and white wire and some wire nuts but it makes for a neater installation. If an outlet connection gets loose, it doesn't affect the whole circuit.
A converted bedroom to an office wouldn't need GFCI protection but you'd add them the same way for a single outlet protection and would use the load terminals for a daisy chain protection.