The Bomberman series is largely the same throught the years and has only changed for weird platformers, racers and stuff.
Easily the best one is Saturn Bomberman, but if you cannot get it, Mega Bomberman on Genesis or Bomberman 94 on TurboGrafx-16 is also good (they are basically the same game), the later of which is on the Virtual Console. There is also a competent set of games on Game Boy Advance, but I forget their names. All of these games have a story mode, with bosses and stuff, something much of the series lacks for whatever reason.
Most of the rest of the games are largely the same, in the cases of stuff like Bomberman Online (Dreamcast) and many releases after, actually removed many features, like the ridable animals. Fans argue that these ruin the games, but as far as I know, they are optional in like every game they are in.
Any other normal release you could find, like Bomberman Party Edition on PlayStation (I think it is on PSN?) are no good at all unless you get a group of friends to play it with. Although when it comes down to it, I would still recommend Saturn Bomberman above all, because it allows you to use a larger amount of people than any in the series that I am aware of otherwise. Also has some cool Hudson characters playable, like Bonk.
Also, it is worth noting that everything besides the ones I recommended sometimes have incredible difficulty spikes. For instance, I have played Bomberman Online for dozens and dozens of hours and have only ever been able to ever defeat ONE out of the six main villains. I am not joking.
Hope this helps.
