There is a main quest which involves capturing and riding a dragon,

Don't get your hopes up though, you won't be riding the dragon a la Drakengard. He will only fast travel you to a new location, and eventually be available as a summon with a certain shout. If you're hoping for a Dragon mount that will carry you all over the place, you'll be dissapointed

No, You Can Summon Odahviing During The Main Questline And Fight Him In Dragonsreach And Trap Him. He Will Take You To Alduin's Portal To Sovngarde, Which Is Located At The Top Of Skuldafn, After That, You Will Only Be Able To Summon Him To Aid You In Battle. Nothing More. You See Yourself Mounted On The Dragon, About 5 - 10 Seconds Later, You See The Loading Screen. So Don't Get All Excited.
Bethesda Decided They Would Be Killjoys.

Well with a new DLC for xbox and pc users, you will be able to ride a Dragon. To bad we dont even have our first DLC. :(

Coming up on the 12 you will be able to if you buy the dlc dragonborn

To use a dragon in battle you can use the shout you get after Riding Odahving (can't remember how to spell it) and for actually riding a dragon you shoud have the dragonborn DLC. It's 50% OFF in PSN. just 'til tuesday 02/19 maybe.