Venonat can be found on Routes 12, 13, 14, and 15, as well as in the Berry Forest, Bonding Bridge, and the Safari Zone. Venomoth can be found in the Safari Zone and in the Berry Forest.
Venonat is a rather interesting Pokémon, being a Bug/Poison type but learning a ton of Psychic moves leveling up. While it doesn't have very good stats as a Venonat, it gets a rather large stat increase when it evolves. Unfortunately, it has sub-par stats towards the late game and it learns all its best moves really late. It also has a lame typing for the late game, so why even use it? For one, it has decent Special Attack and Speed stats, so if that's what you need then Venomoth can offer it. It also learns a few moves from TM and is a suitable pseudo-Psychic-type if you don't have an Alakazam or Hypno, and it learns quite a few status moves. Its abilities (Compoundeyes as a Venonat; Shield Dust as Venomoth) are both pretty good. The best part is, you have several opportunities to get it, and you can even get it as a fully-evolved Venomoth if you're lucky! While better options exist, Venomoth is a decent Pokémon and is definitely capable of making it through the Elite Four.
Gym #5 - Koga (Fuchsia City, Poison-type): If it hasn't evolved, keep it out even if it has Confusion because it's too weak. If it evolves, it might be able to do some damage but be careful of SelfDestruct. Muk will probably be doing more damage to Venomoth unless you picked up the Psychic TM from Mr. Psychic in Saffron. It's probably a good idea to keep Venonat/moth out if you don't have Psychic.
Fighting Dojo (Saffron City, Fighting-type): HA. You have a double resistance to everything in here, so you can spam super-effective Psybeams and Psychics all day.
Rival (Silph Co.): If it still hasn't evolved (which it should have unless you're underleveled), keep it away from any battle. If you have evolved, then things are different. Pidgeot and Fire-types will kill you. Blastoise might fall to a few Giga Drains, but Gyarados will be extremely dangerous. Exeggcute is bulky but offensively pathetic. It also has a double weakness to your STAB. Alakazam only has Future Sight, but Venomoth, again, likely has nothing to hit it, unless you kept Leech Life.
Giovanni (Silph Co., Ground-type): Venomoth might be able to do some damage to Rhyhorn if it has Giga Drain or Psychic. Kangaskhan is too risky because of its bulk and Mega Punch. Nidorino will fall to a few Psychic attacks. Nidoqueen might, but it's really bulky, so it'll take a while to defeat.
Gym #6 - Sabrina (Saffron City, Psychic-type): It would be alright if this was Gen 4 and Venomoth had Signal Beam right about now, but the best move you could POSSIBLY carry against them right now is... Leech Life. Also, STAB Psychic attacks. So don't. However, you could have a Venomoth mirror match if you so desire, but be very careful.
Gym #7 - Blaine (Cinnabar Island, Fire-type): If you want a dead Venomoth, go right ahead.
Gym #8 - Giovanni (Viridian City, Ground-type): Don't let Venomoth handle much here unless it carries Giga Drain. Also, if you have Psychic, then you can handle the Nidos well. However, if you don't have either, it's not a good idea.
Rival (Route 22, pre-Elite Four): Pretty much the same here, except you can actually handle Alakazam if you have Silver Wind and enough levels to actually be able to outspeed it. Rhyhorn will fall to Psychic or Giga Drain.
Elite Four Lorelei (Indigo Plateau, Ice-type): Giga Drain will hit almost everything super-effectively, but only Cloyster will be taking a lot of damage from it. If Venomoth has Silver Wind, then use that on Jynx because it doesn't actually have any Psychic moves.
Elite Four Bruno (Indigo Plateau, Fighting-type): Venomoth does a superb job here. It will at worst 2HKO Bruno's Fighting-types (you might get OHKOs if you grinded a ton and are extremely lucky), but be careful of Rock moves. The Onixes will fall to a Giga Drain each, or two Psychics.
Elite Four Agatha (Indigo Plateau, Poison-type): Psychic spam = Win. However, one Gengar has Hypnosis and Dream Eater, which will kill Venomoth. You have the unlimited-use PokéFlute. Use it. Also, be careful of Golbat.
Elite Four Lance (Indigo Plateau, Dragon-type): Venomoth may have a reputation among the Twitch Plays Pokémon community as a Dragon Slayer, but don't let that fool you. Don't use it here. This isn't Gen I. Everything will be destroying your poor Venomoth.
Champion Rival (Indigo Plateau): Same old, same old. Venusaur will be destroyed, Exeggutor will fall to Silver Wind. Alakazam now actually has Psychic, so I'll let you be the judge of that. (Hint: Don't.) Blastoise MIGHT fall to a few Giga Drains. Gyarados will be too powerful and is neutral to Psychic and Giga Drain and has massive special bulk. Rhydon dies to a Giga Drain, but don't use Psychic because unless you get a crit it won't kill. Charizard/Arcanine will rip your Venomoth to shreds. Oh, and Pidgeot's still a thing, and it can still kill. And it will.
Post-Game: Thinking about using this against Mewtwo? Think again.
Depending on where and when you get it, it can start out with four of the following moves: Tackle, Disable, Foresight, Supersonic, Confusion, and PoisonPowder. They're useful for the time being, but all will most definitely be dropped. The next move it gets is Leech Life, but it's nearly useless and shouldn't even be considered. Seriously, even Tackle is more powerful. Stun Spore is next, which could be used alongside PoisonPowder or could replace it, whichever you prefer. It's going to be dropped anyway. Right when it evolves it gets... Gust. Which honestly is probably better than a few things you have on it but should definitely be dropped ASAP. Psybeam is your next move which is great until you get Psychic. The penultimate move is Sleep Powder, which should certainly be part of its final moveset. Its final level-up move is Psychic, which is a great move but unless you forgot to pick it up or taught it to someone else, you probably already had it on there. You can also go to the Sevii Islands and relearn Silver Wind if you have enough mushrooms.
Unfortunately, it doesn't get much better with TMs and HMs. The best two moves are Psychic and Giga Drain, but let's weigh some things here. First off, it gets Psychic leveling up, so you might want to teach that to someone else. Second, Giga Drain has underwhelming base power and terrible PP. Unfortunately, your other options are hardly worth considering. Physical moves suck because of its terrible Attack. Hidden Power is interesting, but remember that about half the time it'll suck. SolarBeam could be interesting with Sunny Day support, but do you really want to waste turns and moveslots with that? Venomoth isn't bulky enough to set up. Double Team, Protect, Toxic, and Substitute are fun, but rather lame because hax should never be relied on and Venomoth doesn't stall.
Recommended moveset: Psychic, Sleep Powder, and any two between Giga Drain / Hidden Power / Silver Wind
Minimum stats are calculated with 0 EVs, IVs of 0, and a hindering nature, if applicable.
Maximum stats are calculated with 252 EVs, IVs of 31, and a helpful nature, if applicable.
Venonat Line Ratings
What Nature do I want? Modest is the best, assuming you're not running Silver Wind. If so, any other nature that raises Special Attack should be fine. Lowering Special Attack and Speed are terrible because those are Venomoth's only usable stats.
At what point in the game should I be evolved? Probably not long after getting it. Go grind it up before using the thing in important battles.
How good is the Venonat line in a Nuzlocke? While it may not seem very good, it's actually pretty decent. The Psychic-type moves it learns are very helpful, and it's more physically bulky than any Psychic-type besides Hypno (not that it should be taking hits, though). It also has a better Sleep-inducing move than the others. Basically, if you like Butterfree, you'll love Venomoth.