Well, ignoring legendary, mythical, or UB pokémon (all of which are nigh unobtainable), most dragon-types are rare and strong, so the easiest way to answer this would be to simply advise you to consider quite literally any dragon-type in existence. In fact, dragon tamers are a class of trainer who set out to do specifically and explicitly exactly what you want to do; they just settle on dragon-types because those are the most notoriously rare and strong. (That is, they settle on this type for this reason, if not because of tradition or something according to their culture.)
Fire is a bit more varied in terms of what you can and can’t train according to the criteria you’ve set. In terms of rare and strong pokémon, you may wish to consider female salandit/salazzle, which are both wily and deadly but can only be found in the volcanic and tropical areas of Alola. Then you have larvesta, which evolves into volcarona (a fire/bug-type that’s so powerful legend says its heat rivals that of the sun), which is difficult to train because of the fragility of its base form, not to mention difficult to find because it’s only native to very remote spots of Unova and extremely difficult-to-find islands off the coast of mainland Hoenn. While you’re in the Unovan desert searching for volcarona nests (and thus larvesta hatchlings), you may also come across another candidate in the form of darumaka/darmanitan, who possesses potent psychic abilities under certain circumstances.
Then you have Alolan marowak, which can evolve from practically any cubone if trained on Alola only, requiring you to brave the rugged wilderness of the island region. Finally, there are also the seven fire-type starters, although these are specifically bred to be easy to work with, so while they may be strong and rare, they might not be what you’re looking for if you want a challenge.
Personally, though, I think that if you’re an experienced trainer looking for a challenge, what you might wish to do is train a pokémon not known for being powerful or rare. Maintaining a healthy battling record is, from what I understand anyway, fairly easy if you train pokémon that are already powerful from the outset, such as charizard, volcarona, dragonite, and so forth. It’s far more impressive if you manage to succeed on the battlefield with, say, a camerupt, simisear, talonflame, or torkoal. As Karen of the Elite Four says, it’s not about strong or weak pokémon but instead which pokémon resonate with you the most. Pick a pokémon you stumble across on your journey and challenge yourself to train it until you can bring out its inner strength, even if it’s, say, a slowpoke.
Best of luck, anonymous!
The Pokédex According to Bill 