Fire
There are four non-starter fully evolved Fire types (Flareon, Rapidash, Houndoom, Magmortar), two of which are not good/not easily accessible (Flareon, Magmortar).
Rapidash
Rapidash is pretty good. You get Ponyta at a decent point in the game, and Rapidash is very fast and has a solid Attack stat. It is frail, however, and it takes a while to level up. The only physical Fire move it gets until Level 56 is Flame Wheel.
Houndoom
Houndoom is pretty well-rounded offensively, with 90 Attack, 110 Special Attack, and 95 Speed. Houndour evolves fairly early, and it's a nice asset for any team. It is quite frail, however, so be warned.
Water
There are fifteen non-starter, non-legendary fully evolved Water types (Bibarel, Gyarados, Golduck, Floatzel, Gastrodon, Seaking, Whiscash, Quagsire, Pelipper, Azumarill, Octillery, Lumineon, Tentacruel, Milotic, and Mantine). Of these, Bibarel, Gyarados, Floatzel, and Gastrodon are the best possible choices.
Bibarel
A very nice HM Slave, with access to Cut, Surf, Strength, Rock smash, Waterfall, and Rock Climb. It's easily available, though it is weak, and not good in battle.
Gyarados
A staple of many teams throughout the regions, and Sinnoh is no exception. It's a beast, with 125 Attack and a workable Speed stat. You'll have to suffer with Magikarp for a bit, but it's well worth it.
Floatzel
Debatably better than Gyarados, with fewer weaknesses and a better first stage. It's faster, has a very solid Attack stat, and learns a variety of good moves. It's frail, but that hardly matters when it OHKOs everything.
Gastrodon
Gastrodon is the best defensive Water type in the game. It's only contested with Quagsire, but you get Quagsire at a later point in the game. Gastrodon is bulky and a decent attacker, but it doesn't have the best of movepools. It's also pitifully slow.
For in-game purposes, the best Fire type is probably Rapidash, due to ease of access and damage output. The best Water type is probably Floatzel, due to it having fewer weaknesses than Gyarados, and it being almost as powerful offensively. Gyarados, Bibarel, and Gastrodon are all very good as well, however.
Hope I helped!