Great question!
If your market is really strong and you can afford turnover, you could do both at the same time. If you need the income stream of current residents, you may want to implement one or the other (rent bump or charge utilities) at a time for current residents, but you could still implement both the higher rate and start charging for utilities on new move-ins.
Communicate openly with your current residents - if you plan to do one of these at a time, you may want to give them realistic expectations that the following lease they'll have the other, so they can plan their finances accordingly or give you plenty of notice so you can plan.
As for staging the renos, I know a lot of companies go in and do all at once, but I've also seen lots of chaos doing it that way. When given a choice, I like to handle renos at "naturally occurring" turns, which means in most of my markets, roughly 40-50% are completed in year 1, another 25-35% in year 2, and the balance in year 3. Typically that means staff isn't overwhelmed, it's easier to plan and implement, and most vendors now will "box up" your standard reno items and deliver as you need them.