Availability: Early-game (Viridian Forest, level 3-5 (R) or level 3 (B)).
Typing: Bug / Poison makes it good for Misty, Erika, Koga, and Giovanni's Poison-types. It's risky for Sabrina and neutral elsewhere.
Stats: Beedrill's HP, Attack, and Speed are all roughly average, with its Defense and Special being poor.
Movepool: Beedrill struggles with Poison Sting and later Fury Attack until it learns its main STAB move in Twineedle at level 20 (to which Pin Missile can serve as an alternative at level 32). Agility at level 35 and Swords Dance via TM help Beedrill sweep, with Double-Edge serving as Normal-type coverage.
Major Battles: Beedrill loses to Brock but sweeps Misty and Erika easily while being okay for Lt. Surge. Beedrill loses to Blaine, but is good for Koga and Sabrina if it can set up, though it must be wary of Psychic-type moves from the latter. While being decent for Giovanni's Poison-types, it will generally struggle to set up at the Pokémon League due to lacking stats.
Additional Comments: Beedrill is the only Bug-type with proper STAB in Twineedle and is quite good for roughly half the game thanks to the ubiquity of types weak to Bug-type moves. However, it's average ultimately due to its stats falling off and it relying on setup to sweep later on. Weedle is much more common in Red than in Blue.
Beedrill is pretty solid for the early to mid-game, and can put in some work past that, but it ultimately starts falling off when you get access to better Pokemon. It's not the best Pokemon for Sabrina -- physical neutral attacks can do a lot better, due to Twineedle's low damage output and Beedrill's Poison typing -- but it's still decent.