Ryazan's local small airport Turlatovo doesn't operate regular flights and is being used for small aircraft and training. So, the closest air terminals are Sheremetyevo, Domodedovo and Vnukovo of Moscow. From those airports Ryazan can be accessed by train, by bus, or by car/taxi.
By express train from Kazansky Rail Terminal (Казанский вокзал) in Moscow. The ride is a little less than 3 hours.
By bus from Vykhino (Выхино) bus station. The buses leave every hour.
By car / taxi
Ryazan is about 200 km from Moscow by M5 federal road. As elsewhere in Russia, the road conditions differ greatly from a 4-lines highway to narrow bottle-necks with usual traffic-jams. Depending on traffic situation the way from Moscow may take from 2 to 6 hours (beware Friday evenings and Saturday mornings in the summer).
You can get around the city by taking a minibus, a taxi or a bus. The public transport system is represented by ancient trolleybuses which are pretty slow.
The beautiful old Kremlin in the heart of the city, many historical museums inside, on the picturesque bank of the river. http://ryazankreml.ru/
Ivan Pavlov's Museum (Pavlov was a famous physiologist, Nobel-prize winner)
Art Museum. Good collection in an old beautiful building. A lot of good paintings, not typical for provintial museums in Russia.
Drama Theater - beautiful old building
Puppet Theater - prize winning director Valeriy Shadskiy
Take a 3-hour boat trip from the Kremlin embarkment along Oka-river and back. With Russian pop-music and spontaneous dances on the top deck.
Visit bee and honey museum and horse-breeding institute in Rybnoe.
Hotel Lovech. Next to the train-station Ryazan-2. () The biggest hotel in the city, but not in the very center. You'll have to take a taxi to the center (about $3 to the Kremlin) Renovated in 2005. wi-fi. Staff may be expected to speak English.
Priokskaya  Cosy small hotel right next to the Kremlin on a quiet street in the center. Bar and billiards on the 1st floor. Staff may be expected to speak English.
Konstantinovo - the estate and museum of Sergey Yesenin, a famous Russian poet. Great views of Oka river and beautiful rural scenery. About 50 km from Ryazan via the town of Rybnoe.