Of the consumer-level ~55-300 range, 70-300 VR is considered cream of the crop. I've been doing extensive homework when I was on the market for this kind of zoom and this was the best bang-for-buck without breaking the bank.
As far as lenses, 5100 will take ANY Nikon-F mount lens period. You will only get metering with CPU lenses, and auto-focus with AF-S marked ones.
That said, don't be afraid to venture into the vintage territory and explore some of the past lenses since some can really beat lots of the lower end ones today. They'd also teach you how to shoot "film" since you won't have any metering available, but that's why you have Playback button in the digital age.