Well....this may involve using your precious tms....but one of your main problems is levels.
I'll suggest pokemon levels at the minimum needed to have the moves I list, but don't concentrate on the levels too much as long as they're all lv.50-60, it'll be fine.
Try this:
Lv.63
Ice Punch (Rock Slide) / Crunch / Waterfall / Superpower
Superpower is a level-up move at lv.63, so you're nearly there. Ice Fang is fine, just Ice Punch is better, but you'll need to go back to Johto and get some BP in the Battle Frontier to get it, so if you cba, teach it Rock Slide instead, as they have very similar coverage, and Waterfall covers the ground super-effective hit you'd miss out on.
Lv.60
Shadow Ball / Thunderbolt / Focus Blast / Hidden Power
You get brilliant coverage with that, and Thunderbolt is more accurate than Thunder. Hidden Power is good if it's Ice or Fire..if not, stick to Dark Pulse in the last slot.
Lv.60
Cut / Leaf Blade / Sludge Bomb / Leaf Storm (Sleep Powder)
You don't need Flash again once you're in Kanto, as far as I can remember, so you can ditch that. Cut is useful, so I'll leave that. Leaf Blade and Sludge Bomb give dual STAB, and Leaf Storm is powerful, but has a negative secondary effect, which will render Sludge Bomb useless, leaving you with only Leaf Blade to use, so Sleep Powder is an option here.
Lv.60
Fly / U-Turn / Return / Quick Attack
When you battle Blue, you'll find out just how good Return is, trust me, it's worth it. U-Turn is about all Pidgeot has in terms of type coverage, and Quick Attack can pick off weakened enemies.
Lv.60
Air Slash / Aura Sphere / Extrasensory / Wish
Once you have 16 badges, go to the National Park and use Rock Climb to find a Shiny Stone for evolving your Togetic into Togekiss. Aura Sphere and Air Slash are brilliant moves, Extrasensory is too. Wish is good, as it can be used to heal either the user, or whoever switches in next turn.
Lv.60
Stone Edge / Iron Tail / Crunch / Strength
Rock Smash isn't needed, so scrap it. Steelix has pretty good options available, those 3 above are the best.
To be honest, you have no fire in your team (a problem mine had too), so I'd suggest capturing a Houndour, as it's pretty good, or a Growlithe, then try one of these, probably instead of Pidgeot:
Nasty Plot / Flamethrower / Dark Pulse / Overheat
Overheat is good, but has bad side-effects. Therefore, you can either use Nasty Plot to cancel it out, or just use Nasty Plot and sweep with Flamethrower and Dark Pulse. Shadow Ball, Solarbeam and Iron Tail are also options.
OR
Extremespeed / Dragon Pulse / Flamethrower / Dig (Thunder Fang)
Arcanine is much better coverage-wise, Dig is useful, as it is powerful and can be used to escape caves, but if you don't want it, Thunder Fang enables you to hit water-types!
So, that's my two cents on the piece, but it's up to you really. Anything can win through eventually, so you don't necessarily need a fire-type..