Having been with friends and family who don't like horses, I've never tried it, but Swaledale has several stables and the one most known and closest to Reeth is probably the Arkle Moor Riding Centre, just outside Langthwaite in Arkengarthdale.
Reeth is along the famous Coast to Coast Walk which you can read much more about in the webpages below. This means that lots of B&Bs and hotels cater for walkers with special services such as luggage transport, storage rooms and other things that walkers might need. To do this walk in all its lenght, you need at least two weeks, but around Reeth you can do the Swaledale sections of it, such as walk to Gunnerside (quite a nice and easy walk) and Keld or towards Richmond. You can also go hiking up Arkengarthdale. The national park centre has several books for walkers but if you want to do advance planning, again check the websites for literature and maps. Remember that the weather can change rapidly in the hills here whatever the season and that nice weather in Reeth does not necessarily mean nice weather in the next village so bring waterproofs and good boots.
Every year in Reeth the Swaledale Marathon is held.It is a 40 k run or walk within 10 hours.The route is very hilly & most of it is off road.The event is held in June & it is open to the first 500 applicants.It is a great day out with fantastic views if you have time to enjoy them.