I think the idea of analysing first gym leaders is particularly interesting. There are trends that can be observed that are a little strange. The typings of the first gym leaders in different games in the series are Rock-, Flying-, the type that's SE on your starter (FWG), Normal-, Bug-, Normal- or Normal-/Dark-, Grass-Type. For me as someone who really doesn't like Totem Battles, this means that there's only one contender for the best first gym battle in the series.
Best: Cheren (BW2)
How is Cheren the only first gym leader to use a typing that doesn't advantage or disadvantage certain starters over the others? Not only that, but they intentionally put his levels just low enough even in Challenge Mode to mean that so long as the player isn't over-levelling none of the starters will have evolved, so Pignite's secondary Fighting-Type isn't even a factor. It's the start of the game -- players have the option of Water-, Grass-, Flying-, Normal-, Fire- (starter only except Sinnoh and Kalos iirc), Bug-, and sometimes Dark- (Hoenn & BW1 only iirc) and Fighting-Type Pokémon (Kanto, Sinnoh & BW2 only iirc). Players don't have a lot of team options and in some cases choosing a certain starter all but necessitates a certain wild Pokémon: if you choose Fire-Type in any of the regions with a Rock-Type first gym leader you're essentially forced into using Mankey in Kanto, Shroomish / Lotad | Seedot (if you use Bullet Seed TM) / Wingull in Hoenn, and Machop / Budew / Psyduck in Sinnoh. If you choose Chikorita in Johto you have to use a Geodude or throw a full team of 6 Normal-Type Pokémon at Falkner, and in BW1 you're literally forced into receiving and then intuitively using an elemental monkey. Admittedly what
Yung Dramps said about Viola is true and something I hadn't noticed before, so the type matchup isn't so clear-cut there. However, using one starter still gives a big advantage over the others -- it's just not in an intuitive way, so Chespin is the best matchup against the Bug-type gym leader whereas Fennekin and Froakie struggle a little and are forced to use a Flying-Type, or muscle through the Surskit with Normal-Type moves before using a Litleo on the Vivillon.
It's a very strange issue because the series has options available to it. Normal-, Fighting-, Psychic-, Dragon-, Ghost-, and Dark-Type are all viable options for early game gym leaders that wouldn't advantage one of the Fire-, Water- or Grass-Type starters over their alternatives, assuming the base forms of all of the starters are pure type. It seems like intuitively better game design to use any of these types as the first gym leaders for that reason, but in all 7 generations featuring gym leaders, only one actually does so. The only reason I can think of is that for less experienced players they can choose the starter that has the easiest time against the first boss battle, and the more experienced players can choose a challenge route if they like. However, this only works if players have knowledge of what type the first leader is which is never presented in-game before the starter is chosen, and by the time players are playing through the game for a second time none of them should need to take the easy road because they've understood the game enough to take down the Champion already. Furthermore, if that truly is the logic behind this design, Cheren further proves that it's not necessary to wrap the starters up in providing an easy route. Riolu can be found rarely in Floccessy Ranch and comes knowing Fighting-Type moves. If you're struggling against Cheren, you can catch a Riolu and beat him without breaking a sweat.