Socio-political systems are technologies. They are developed and refined as people navigate their lives and experiences within the frameworks of their culture and environment. Much like dragons, the vikings (and other raiders) of real life presented a threat that small communities could not effectively deal with on their own. Feudal society is almost a direct outgrowth of these kinds of pressures upon agrarian city-states such as those described. Individual city-states do not have the economic power to fund the kind of on-demand skilled military force that is the necessary response to a dragon incursion. Through collaboration and subordination, valuable regions can be assured greater protection through the redistribution of material resources perhaps more plentiful elsewhere at minimal cost. The existence of a trained standing military force (knights) which, due to feudal arrangements, is distributed thoroughly throughout civilian populations is an effective and specialized response against spontaneous small raiding excursions of the sort that you seem to be imagining dragons making. In cases wherein dragon occupation is achieved-- a settlement or region being dominated and its local military proving insufficient-- the ultimate authority of a King allows for the potential to raise the necessary forces and to ensure those forces are sourced in a politically-sustainable manner. Regions that are occupied may be left occupied when the cost of liberating them is deemed to be prohibitive, but the cost will not be prohibitive in all cases.
In terms of the actual slaying, dragons of the kind you seem to be assuming are normal tend to lair in defensible underground locations-- high mountain caves and such. This use of terrain allows a Dragon put on the defensive to effectively negate any numerical advantage human attackers might have, within genre (outside of genre the humans might maintain a siege or send so many soldiers that the human dies of exhaustion, but both options wouldn't work with, for example, Beowulf's bane). Furthermore, the jets-of-flame manner of attacking is significantly more effective against lightly armored rank and file infantry than heavily armored elite troops. Thus individuals or small groups will be the desired norm for the Dragonslayer.
Dragons seem to generally be able to sneak from place to place fairly quickly and without much time to respond. Potential dragonslayers must be able to travel quickly in order to have any hope of chasing down and outmaneuvering a dragon, or they must successfully track a dragon down to its lair, which is not an easy task. Dragons, as the physical incarnation of greed, hoard treasure. Dragonslaying is thus extremely profitable, and-- as dragons are quite rare and take a long time to come to be (via reproduction or spontaneous generation or curses or however your dragons come to be)-- dragons will thus quickly become very rare and reportings of a dragon will be responded to quickly by a number of would-be slayers as the dragonslaying industry progresses over time. Thus in the beginning of successful anti-dragon warfare horses are very much desired, and as the industry progresses they become almost necessary. Horses are very expensive in medieval societies, so this greatly limits who can enter dragonslaying as a profession. The spoils of dragonslaying are, as previously mentioned, great, however, so successful dragonslayers will have no trouble maintaining their status barring political action.
Plate armor is probably the most effective form of armor against dragons. The full body covering with internal padding should render one practically invulnerable to its flame except for direct attacks through the visor. The issue then becomes dealing with the dragon's thick hide and multifarious sharp body parts, which is largely a contest of skill, strength, and weapon quality.
The social generation of knights in shining armor is thus a reasonable solution to dragonslaying, as well as a classic one.
Note that the above assumes the standard, flightless dragon, such as that found throughout Nordic myth, rather than the winged versions common in modern Tolkienian fantasy (or Tolkien's inspiration, Beowulf's bane). Flighted dragons nonetheless seem to dwell in the same sort of places, which obviously prevents flight while in its lair, so the method of dealing with them would rely exclusively on hunting down and killing them proactively as killing them 'in the field' would be technologically infeasible until ballista technology is recovered. As the Renaissance dawns you could make use of such equipment as the Polybolos, rendering wealthy settlements immune to the wrath of dragons.
