Lapras
High Tier
Availability: Can be caught immediately after Morty, requiring Surf. Comes at a respectable level 20 (or 25, I forget), but can only be caught on Fridays in Union Cave.
Stats: Excellent HP, reasonable offences and defences, below-average speed.
Moves: Give it Surf immediately. Gets Ice Beam naturally, also has access to Body Slam, Confuse Ray, Thunderbolt, Psychic, Solarbeam, Earthquake, Strength and Whirlpool for HMs, and more.
Power: STAB Surf and awesome coverage off solid offences. What more do you need?
Type: Water is wonderful, but the Ice type hinders it, offering no benefits apart from STAB Ice Beam. However, Rock isn't a problem, Steel and Fighting are quite rare, slow Grass is generally covered by Ice Beam, leaving Electric as the main problem.
Pre-Kanto Matchups:
Chuck - Chuck wins easily, don't bother.
Jasmine - Steelix has Sunny Day, but you'll win against it easily. You can probably one-shot Magnemites with Surf too, if you feel like taking a risk.
Pryce - If Lapras has Thunderbolt (via tradebacks), this is curbstomped. If not, it's a bit of a war of attrition with Body Slam. You WILL come out on top.
Team Rocket - Vulnerable to Poison, but bulky enough to handle it, and powerful enough to deal with most mons, especially the Executive's Houndoom.
Clair - Watch out for the Dragonair's Thunderbolts. Ice Beam deals with them though, and Lapras is one of the few mons who can reliably stand up to Kingdra.
Will - Lapras can easily beat the Xatu and Jynx. Slowbro is a painful war of attrition without Thunderbolt, and Exeggutor is best avoided.
Koga - Lapras should win this pretty handily.
Bruno - Onixes are easy, but the Hitmons need to be avoided.
Karen - Murkrow and Houndoom are easy prey (watch for Sunny Day/Solarbeam though), Gengar is 2HKOd by Surf, Umbreon can't do shit, and Vileplume is slow enough to be Ice Beamed.
Lance - Lapras positively shines... Unless she gets Thunder'd.
Kanto matchups:
Brock - Lol
Misty - Lapras handles everything except Quagsire well (unless she has Solarbeam).
Lt. Surge - Don't kid yourself
Erika - Can go either way, depending on how fast Lapras is
Sabrina - Bulky enough to sponge Psychic and do reasonable damage back.
Janine - Lol
Blaine - Lol
Blue - Lapras can probably deal with Rhydon, Arcanine, Pidgeot and possibly Exeggutor. Gyarados if it has Thunderbolt too, and maybe Alakazam on a good day. So yeah, Lapras > Blue.
Red - Blastoise and Charizard are easily handled.
Avoiding Top because it comes mid-game and needs TMs/tradebacks for a lot of its best moves.