I live in Harrogate, but am not sure about your first question as I normally drive. I'll investigate and get back to you, as I assume you want to take a train and not hire a car?
I can answer the second question! There is a train service running from Harrogate to York. Think the trains run hourly during the day, more frequently at peak times - this website will help with train times (might also help your first question too).
Harrogate to York takes about 40 minutes by train - pretty journey too. York Station is central to the city.
There is no direct bus service from Harrogate to York.
However, there is a direct bus from Harrogate to Leeds. Leeds is FANTASTIC for shopping, with an amazing range of shops including Harvey Nicks. If shopping doesn't interest you, then half way to Leeds is Harewood House (the bus stops literally at the gates of the house). Beautiful place, well worth a visit. Buses run every 20 minutes most of the day to Leeds, and all pass Harewood. See websites below for details.
Shout if you have any more questions about the area.
You can take the train from Manchester Airport to Harrogate, but you'll have to change trains at Leeds. The journey in total will take approx 2.5 hours.
Manchester Airport to Leeds on Transpennine Express.
Leeds to Harrogate on Northern Rail.
You can plan your journey, and get specific times, at www.nationalrail.co.uk
I also meant to say that I used to live in Harrogate, and would fly from Manchester Airport to the US - when I did, I used a taxi service, but can't remember which one. However, a list of private transfer companies is available on this website (scroll down to Manchester Airport):