Among series with more than two games, Uncharted and Metal Gear Solid come to mind. Uncharted probably doesn't need a ton of explaining, but Metal Gear Solid is probably more controversial. Personally I enjoy its story the most, perhaps after MGS1 though. What puts it ahead of MGS1 is everything else though. I also like its gameplay more than e.g. MGS3's, which is a popular favourite. MGS5 might be ahead in many ways in gameplay though, but it's a much more flawed game than MGS2.

For series with only two games, Left 4 Dead comes to mind. I think few would pick the first game over the second, so this one probably doesn't need explanations. Cities: Skylines is a more controversial pick though, because it is very much a flawed game. However, as someone who doesn't play with a ton of mods, the improvements just make it so much more fun to play than the first one.

Mega Man 2
Dark Souls II
Street Fighter II
Gargoyle's Quest II
Star Fox 64
Paper Mario: The Thousand-Year Door
Kingdom Hearts II
Octopath Traveler II
SteamWorld Dig 2
Yooka-Laylee And The Impossible Lair
Maximo VS Army Of Zin
Alice: Madness Returns
The Last Of Us: Part II
Uncharted 2: Among Thieves
Daymare: 1994 Sandcastle
Silent Hill 2 (remake)
Far Cry 2
Guitar Hero II
Rock Band 2
Half-Life 2
Zillion II: The Tri Formation
Wario Land II

Some of you needs to play Street Figher III 3rd Strike. That is the best in the franchise by a large margin. SFII is the most important tho but go back to it now. It's bare bones.

Thats why I like it. That its bare bones. Not a big fighting game fan.

I also forgot about Dark Chronicle (Dark Cloud II)

Diablo II
Age of Empires II
Portal II
Half-life II
Torchlight II
The sims II
Sonic II
Donkey Kong II
Alan Wake II
Uncharted II
Mass effect II
Death standing II
Titan Fall II
Destiny II

Dead Space 2.

Mass Effect 2. - Xbox 360/Playstation 3/PC.

* Improved visuals.
* Improved and refined game mechanics.
* Much tighter story.
* More nuance in the characterization of characters.
* Vastly superior combat system.


**************

Lufia 2. - Super Nintendo.

* Better visuals and Mode 7 effects on SNES.
* Much better sound track and audio effects.
* Despite being a Prequel, the story is more coherent.
* Ancient Cave.
* Characters have more personality.
* Better puzzles.


**************

Donkey Kong Country 2. - Super Nintendo.

* Pirate thematics dripped with charm and atmosphere.
* Vastly improved Audio tracks, utilizing the SNES's audio capabilities to the fullest.
* Improved SGI pre-rendered sprites, with more detail.
* More gameplay mechanics. (Animals)
* More collectables. (Coins)
* More and better scan-line scaling effects for some ridiculously good looking 3D-esque backgrounds.
* Tighter controls and general gameplay flow.

F-Zero X (I know, the Satellaview GP "BS" games exist, but those were kinda like a primitive form of DLC, so I'm gonna cheat a bit) was a fantastic racer with style and attitude to spare that still plays great nearly 30 years later, which is something not many 3D games of that era can boast.

GX was amazing too, but just a tad too unforgiving for my tastes.

Batman Arkham City
Mega Man 2
Halo 2
TimeSplitters 2
Sonic the Hedgehog 2 (well Generations would eventually top it, but that is in 3D, 2 is still the best pure 2D Sonic)

Half-Life 2 is a contender, but it is basicly a coin toss between that and the original. I also haven’t played Alyx. Portal 2 trumps 1, but it is to small a series to really be considered.

TIE Fighter
Carmageddon II
The Sims
Pokémon Gold and Silver
Indiana Jones and the Fate of Atlantis (franchise is wider, but I'd call this second in the series after Last Crusade and on the other hand later games, aside Infernal Machine, aren't really part of same series - nor have I played them)