Hi Marisa -
You could probably fly into London Heathrow and travel north from there by train to Leeds. Or Manchester if flights are economic and/or available. Again from Manchester your destination would initially be Leeds by Rail.
Leeds is a good access point for Howarth. From there take the train (or bus) to Keighley and from there, either connect by bus or more usually the privately run Keighley and Worth Valley Railway to Howarth, dependent on what times work best for you. Possibly here is a good place to advise that ‘Keighley’ is pronounced ‘Keeth-ly’! From the railway station at Howarth to the Parsonage involves a long, steep, cobbled climb up Howarth Main Street, so hopefully you are fairly mobile! If walking is going to be difficult I suggest a taxi to the Parsonage from Keighley.
To reach Fountains Abbey from Leeds take the frequent No. 36 bus via Harrogate to Ripon. On Mondays, Thursdays and Saturdays the 139 bus runs from Ripon to the Vistor Centre at Fountains. On Sundays in summer it’s the Dalesbus as linked above. All the bus services are infrequent, so if nothing suits either getting there or getting back, a taxi will be just about your only option. However, if you are OK with walking there’s a walk published by the National Trust which details a walk to and from Ripon to Fountains. It’s an eight mile round trip so the walk one way will be about four miles. It’s worth going the one way at least via the Studley Royal Deer Park, splendid walk in good weather and if possible extend the walk a little to St. Mary’s Church, which is also worth a visit if open. Again on the walking front from the Fountains Visitor Centre if you get dropped off there there’s a walk down to the Abbey Ruins on well made paths, but what goes down..... (The abbey is in the valley bottom and the Visitor Centre is on top of the hill above). Once down at the ruins there is extensive walking in and around the ruins to and through the water gardens, Banqueting House, Lake, lower tea room, etc.,
A quick word about where to stay. Leeds is my home town but it can’t be described as tourist central. It is a good transport hub and would be good for Howarth, but you could stay further out in, say, Ilkley, which is slightly more rural, and which has good rail connections from Leeds and connections to Keighley via Shipley, I think. Not too far from Ilkley are Fountains-esque parklands (more walking!) and semi ruined (front half is still a working church and can be visited usually) Priory Church at Bolton Abbey. If you are particularly interested in ecclesiastical ruins from the Dissolution, Leeds has its own in the form of Kirkstall Abbey, a huge ruined Cistercian Abbey (built 1256 I believe), about three miles out from the centre of Leeds.
For Ripon and Fountains, either stay in Ripon or Harrogate - like Bath, Harrogate sprung up as a spa town and has a similar genteel, if undeniably northern, feel. It’s also home to the original Betty’s Tea Room, usually mentioned here with regards to York, which has two. (Harrogate has two nowadays, the second being a little way out of town at the Harlow Carr Gardens).
Buses and trains connect Harrogate to Leeds for onward travel to Chester and Bath by rail.
If visiting Stonehenge I always suggest trying to get on a tour that will also take you to Avebury where you can get up and personal with the stones. The village of Avebury sits within the circle which will give you a sense of the sheer size of the place! Unfortunately Google is your friend here as being a Brit I’ve never taken a tour, having arrived by bus (from Swindon), car, or on foot having completed the Ridgeway long distance trail.
Feel free to PM me if you need any further info or help with travels in Yorkshire. I hope you have a great trip!
Ian