Dodrio doesn't have a lot of options but I'd recommend this:
-Drill Peck
-Steel Wing
-Tri-Attack
-Sunny Day/Mimic
The first three are solid 70-80 power (+STAB for some) attacks that work well with Dodrio's high Attack/Speed stats. Tri-Attack also has a chance to Freeze, Burn, or Paralyze your foe.
Sunny Day can be used to help Arcanine or Venusaur with Fire attacks or Solarbeam. Mimic provides a potentially useful move depending on the battle, especially if it gives you a strong Physical move.
First I want to warn you about catching a Chansey. They're extremely tough to find, not to mention catch (although Dragonair and Tauros are even harder out of the Safari Zone Pokemon in my opinion). My move set recommendation is this:
-Substitute
-Softboiled/Rest
-Toxic
-Seismic Toss
This is THE Chansey set. Accept nothing less. Note: Do not use this against Agatha in the Elite 4, since it can't do anything against Ghost types. Also not against any strong Fighting types. Pretty much everything else is doomed by the original wall o' decay though unless Chansey is poisoned or burned. You can restore more health and status problems with Rest, but Softboiled doesn't put you to sleep, so take your pick.
Venusaur:
-Solarbeam/Leech Seed
-Giga Drain/Earthquake
-Sludge Bomb
-Synthesis/Mimic/Earthquake
If you're running Sunny Day on Dodrio or Arcanine use Solarbeam and Synthesis. Otherwise you're better off with Leech Seed and either Mimic, (which may help Venusaur's very poor variety). Earthquake is always great, especially against Agatha.
Arcanine:
-Flamethrower/Fire Blast
-Extremespeed
-Iron Tail
-Hidden Power/Sunny Day/Bite
This guy has some solid mixed stats. Decent bulk, speed, Attack, and Sp. Attack make him not too far off Mew's equal spread of stats. If only he had Mew's move pool. Hidden Power could be useful if its type and power are good (Grass would be great), Extremespeed offers priority if you run into something with higher speed, Flamethrower and Fire Blast offer power or accuracy, Iron Tail helps cover Rock, and Bite is usable if you don't want Hidden Power or Sunny Day.
Finally Kabutops. His move pool is awful despite his appearance (since he looks like he's be tougher) but high Attack and Defense, along with above-average speed, can be helped with TMs.
-Return
-Surf
-Aerial Ace
-Rock Slide
High Attack enables Rock Slide and Return. Surf is good because despite the low Sp. Attack it is a strong Water-type move with STAB and Kabutops is the only one one your team able to use a water type move other than Water Pulse (and there's no reason to give Chansey Water Pulse since it lacks STAB and its Sp. Attack is even lower than Kabutops'). Aerial Ace hits Grass and Fighting decently. It'd be nice if Kabutops learned Earthquake, but unfortunately it can't.
Mr. Mime can only learn the elemental punches in Emerald, so Ice Punch isn't available. I'd use the following:
-Psychic
-Calm Mind/Meditate
-Thunderbolt/Thunder/Meditate
-Baton Pass/Magical Leaf
Baton Pass is great if used before Mr. Mime is defeated. It gives any status changes to another Pokemon in your party and switches with it. Great for setting up on your foe by boosting stats and sending in someone like Arcanine or Dodrio. Meditate is useful for Arcanine, Kabutops, Dodrio, and Venusaur with Baton Pass while Calm Mind is better for Mr. Mime itself, Venusaur, Arcanine, or Kabutops. If you don't plan on using Baton Pass use Calm Mind, Magical Leaf, and choose between powerful Thunder and accurate Thunderbolt. Psychic is a great move that should be used with or without the Baton Pass set.
And that's it! Sorry if it's a bit long, and hope it helps.