Beedrill

Beedrill lives up to its initial use in the original games flawlessly: an easily obtainable pokemon that's easy to raise and help bolster an early team. Sadly, its best role is nothing but a stepping stone to greater things in the game as any real collecting will yield better poison and bug types and leave Beedrill outclassed. On defense it's got a tiny bit of merit early on due to double-resisting Machamp's signature attacking set, but by that same token its entire moveset minus Elite Moves is hard-walled by Steel types.

The only real reason to raise a Beedrill is for Mega Beedrill.

Best PvE Offensive Moveset

• Bug Bite* and Infestation have roughly similar performance, with Infestation having better damage and Bug Bite better energy gain, though the legacy Bug Bite pulls ahead just slightly. 
• Poison Jab is good when paired with Sludge Bomb, though Sludge Bomb pairs with Bug Bite for Beedrill's highest performance against grass types and neutral targets.
• X-Scissor is fast and furious, giving Beedrill a bug type set.
• Aerial Ace is fairly slow, but if you've just started and are desperate for fighting-type counters, you can leverage Beedrill's double fighting-resistance and use this. However, even many neutral matchups can out-damage Beedrill on this metric.
• Fell Stinger has a higher attack rating than X-Scissor, but it's substantially slower which makes it weaker in practice.
• Beedrill is a bad Bug-type attacker. It's an even worse Ground-type. Leave Drill Run alone.

Best PvE Defensive Moveset

• Infestation outclasses Bug Bite hands down.
• Poison Jab has the advantage of being off-type with Bug Type moves and has good power on defense, but Poison and Bug don't synergize extremely well offensively.
• Because Beedrill is so fragile, Fell Stinger is probably Beedrill's best move on defense, followed by Aerial Ace. Being three-bar, they have the best chance of actually firing on something so frail
• Drill Run* works well for off-type damage, and has outstanding coverage. Its one down-side is the fact that it doesn't come out as frequently as Aerial Ace or X-Scissor.
• Sludge Bomb is one of the best options for pure damage, though it's less reliable due to having fewer bars than Beedrill's other moves.
• X-Scissor is extremely similar to Fell Stinger, but slightly weaker on defense.

When looking at stats, Poison Jab is superior to legacy Bug Bite, though they're different typings and cover slightly different roles. Poison Jab has recently  become a stronger choice than it once was thanks to the presence of Charm users such as Wigglytuff and Clefable. Infestation has the highest gains by a very small margin, but lacks raw damage output. 

X-Scissor and Sludge Bomb gain STAB with Beedrill and solidify its role as a Bug/Poison type. The former costs very little energy to use and can be spammed to bait shields, while the latter packs a harder punch with a higher 1.6 DPE rating. Aerial Ace is an inferior move that doesn't get STAB even if it would offer coverage against Fighting-type Pokemon. The addition of Fell Stinger following the Dec 11, 2019 rebalance is interesting for Beedrill, but isn't very impactful if it doesn't bait a shield due to its pitiful damage output. Beedrill's Community Day move, Drill Run, gives it the much needed coverage against Steel-types that it's STAB moves lack. It's also quite a powerful move.