What models would you suggest that I get to make this more competitive in games I play (normally against SM, CSM, Orks, Dark Eldar, Tau and Eldar)?
Flying Hive Tyrant w/ 2x Twin-linked Brain Leech Devourers. Add the haywire chest flamer if you're feeling frisky.
"Competitive lists" run something like 5 of them.
Maybe some Mucolid Spore Clusters if you need troops or are required to be Battleforged.
Also what would be a good way of running any of the stuff I've got, especially the genestealers, Carnifex and warriors?
If you're keeping with what you got, first off, get a Venomthrope/Zoanthrope box. Build up at least 2 Venomthropes. Learn to use your Venomthropes and your Gants/Gaunts as a screen, providing all your 'Fexes, Warriors, and Tyrant with a 3+ cover save. It drastically increases your durability while you cross the board.
The "best" loadout for the 'fex is still the dual TL-BLDevourer build with Adrenal Glands. That will get you 24 str 6 shots, from the brood, as well as being Str 10 on the charge (and allow you to re-roll charges). A good all-rounder unit, though a bit slow.
The "best" loadout for warriors has been the minimum squad with a barbed strangler to act as objective campers, with Deathspitters an optional upgrade. Personally, I run them in large squads of 5 as melee Warriors (adrenal glands, flesh hooks, no guns, 1 bonesword/lash whip, 1 rending talons), and am pretty happy with it. It's not "competitive", though, but it tears infantry models up. Fleet allows you to get across the board quicker than you think.
Genestealers are a hard unit to use. Most people I see run them as minimum squads just for the Broodlord, with clever use of his spells. I've seen one guy use them effectively in bulk, mostly by keeping them in reserve for at least a couple turns, and generally outflanking. That way he's weakened his opponent with the rest of his army, plus his opponent has had time to disperse some so as not to bring his whole army to bear against them when they arrive. Basically, since Genestealers are troops, he brings them in late to make an assault against a decisive target in the last turns of the game. I wouldn't recommend them for most people.
Personally, I run a list very similar to this (gaunts in the front, warriors in the middle, 'fexes in back). It's a hit and miss list. Generally, if I can get across fast enough, I can typically overwhelm my opponent. Sometimes, they do so much shooting to me in the first couple phases that my list just crumbles, like the tide against a rocky shore. I enjoy the army, because it really has that "carpet of gribblies" feel, even if it is a bit tactically shallow (tactics = push all models forward, cross fingers that enough live to do damage).
Fleet (Adrenal Glands) are the key to my list, because a foot-slogging Tyranid army is sooooo slow. My list would never succeed without Fleet. That and I really rely on symbiotic relationships (like using gaunts as mobile cover saves). Most "competitive" builds offset the lack of speed with Mawlocks and Tyranocytes (Deep Striking) or with Wings (ie: lots of Flyrants). A Walking Tyrant with Hive Guard is a survivable combo, but a Flyrant (with Devourers) is much easier to use, and you'll feel like you get a lot more mileage out of him. Look on bits sites, you can probably find a set of wings cheap, and just stick them on your current Tyrant.
Personally, though, I think you could run test games with what you have. Proxy your zoanthrope as a venomthrope (since they are a similar silhouette), and keep your models around him as a node.
Any questions, feel free to ask. I know I've found something that "works for me", so I tend to get a bit myopic about it. Every other way I've tried to play 'Nids tends to fall apart on me (though I haven't tried 5 flyrants, to be honest), so I've stuck with it. Others will be able to provide other perspectives.