There are so many factors steering you to your most enjoyable stay.
Cost: Top dollars lets you experience the best but for a first stay something mid-range would probably be better. Cheapest might be a bit tough for a first visit.
Location: City, Rural, Thermal district. For a first stay one of the Onsen towns would be a good introduction and there are several en-route Tokyo-Kyoto.
Ryokan style: These range from newish, multistory buildings like hotels to very Japanese establishments with exquisite gardens, exotic thermal baths, kaiseki dinners served in your room and an atmosphere that is 'authentic'
The full experience or with a few concessions to the non-Japanese traveler: Essentially the concessions available are with the seating and the bedding. The typical ryokan has you sleeping on futon laid out on the tatami floor. It is an experience to try it, but for an older westerner getting up from floor level can be a bit of a challenge. Some ryokan have one or two rooms with futon-beds, that is the bedding is raised up on fixed plinths. I would recommend not going for this style even if it is a bit less comfortable because it interferes with the concept of the tatami room that is the main feature of a rypkan stay.

If you let me know your reaction to the above I will come forward with a few suggestions.