I saw Exeggutor and checked up on him and it SUCKS, so of course I'm gonna use him but I need more.
He's not that bad, IIRC. It's bulkier than your standard Psychic, hits decently on the physical side, and quite hard on the special side, and has some decent status moves because it's Grass-type. It suffers from a fairly bad basic movepool, even taking Exeggcute's moves into account, though, and wants TMs and Move Tutors. Things like Psychic, Giga Drain, and Sludge Bomb make good attacks for it, (it can learn Solar Beam naturally, but unless you want to give it Sunny Day...) and if you want you can teach it Explosion via move tutor. A detonating palm tree could be a decent finisher for some matches. Statuswise, Leech Seed can be absolute gold if used right, and Stun Spore or Sleep Powder make catching things easier.
As for other team members... You could get a Ditto and name it SPY... Or a Chansey and name it MEDIC, or a Rhydon (or Snorlax!) and name it HEAVY... Magmar might be a appropriate-looking fire type, since we passed up on a Flamethrower user at the beginning, but you can get a Vulpix much earlier. (I'd have suggested Growlithe, but that's Fire Red-exclusive) I'd also suggest Scyther, because every military team needs a knife nut.
But here's the problem: Virtually everything I've mentioned is found in the Safari Zone or later. You need suggestions for Pokemon to carry you that far in the game. To that end, I'm going to suggest Diglett or Sandslash, Bellsprout, and... I guess something from Mt. Moon: Soldier clearly needs a Clefairy in his employ.