Unfortunately, Squirtle (nor its evolutions) actually learn Ice-type moves by level-up. It can, however, learn Ice Beam and Blizzard through TM. The earlier (and better, IMO) would be Ice Beam, available in the Rocket Game Corner of Celadon City, if I recall correctly.
But by then, you should have access to a ton of Pokemon that would be a bit cheaper to take down that rival of yours. Assuming you're referring to his Bulbasaur, you've got Pidgey/Pidgeotto/Pidgeot, Spearow/Fearow, and Zubat/Golbat, all before you hit Cerulean City and all sporting Flying-type moves that are fully capable of taking down a Grass-type. Then, there's Butterfree, who not only has Flying-type moves but also Confusion, which wreaks havoc on Bulbasaur's Poison-type. Then, on top of that, Rattata/Raticate does neutral damage and blitzes the in-game trainers with Hyper Fang, Jigglypuff and Clefairy can Sing and Pound the crap out of unsuspecting victims, and Paras resists Bulbasaur's Grass-type moves liek whoa and learns Slash at the tender level of 22. And that's just by Cerulean City. So, really, you can probably get by without an Ice-type move.