Since your Typhlosion is already around the same level as Red's Pokemon this answer will focus on it but can still be applied to all of your other Pokemon with a few minor changes.
To beat Red more easily it'd be a good idea for Typhlosion to know moves that are super effective against Red's Pokemon. Here's a list of 4 types that are super effective against his team as well as moves that Typhlosion can learn for each type.
Fire: Flamethrower, Eruption, Fire Punch etc. (super effective against Venusaur)
Electric: Thunder Punch (super effective against Charizard, Blastoise and Lapras)
Fighting: Focus Punch, Focus Blast, Brick Break etc. (super effective against Snorlax)
Ground: Earthquake, Dig, Mud-Slap (super effective against Pikachu)
A full list of the moves Typhlosion can learn in HeartGold can be found here. A list of Red's team in HeartGold can be found here.
Having Typholsion hold an item can also help with the battle. The ones listed below are the ones I'd recommend the most but a full list of items that can be held by Pokemon can be found here.
Expert Belt (Found on Mt. Silver)
> slightly boosts the power of supereffective moves.
Leftovers (Held by Snorlax found on route 11)
> The holder's HP is gradually restored during battle.
It'd also be a good idea to bring as many Full restores and Max Potions as possible to make sure Typhlosion doesn't faint.
The last suggestion I have to sweep Red's team is to level up Typhlosion a few more times considering Red's Pokemon range from level 80 (Lapras) to level 88 (Pikachu) with the starters being level 84 each.
Those are all the suggestions I have so I hope I've helped and good luck with sweeping Red's team.