Taverley (also spelled "Taverly") Dungeon is a big and good place for training, but there isn't a bank close by. Taverley is the village that the dungeon is named after, and it's the closest one you'll find. This dungeon is also called the "Members Dungeon" because it's the first one that many players enter once they become Members.
Here is the outside city and closest city to the dungeon. The red circle is the dungeon and the orange circle is Taverley:
When you climb down the ladder and enter the dungeon you will come across two Agility shortcuts. The first (to your East) is a pipe that requires 70 Agility to crawl through and leads you to the Blue Dragons. The second shortcut is slightly further on and requires 80 Agility. This leads to the Poison Spiders.
The Cauldron of Thunder
As you start to head North from the entrance you will see a gate to your East. If you try to open the gate you will be attacked by two Suits of Armour. These aren't a problem if you click the gate quickly three times. Through the gate is the Cauldron of Thunder which plays a role in the Druidic Ritual Quest. Past here there are two gates that you could go through. One to the East and one the the South. Through the Eastern one is a room full of Bats and through the southern one are Ghosts.
If you didn't choose to go through the gate to the Cauldron of Thunder further to the north you will come to some Chaos Druids. They are level 13 and are often killed for the Herbs they drop so the area that they live in is often busy. If you go South from here you will go past some Black Knights and through a room full of Magic Axes. Through this room are some level 20 Poisonous Scorpions and some Chaos Dwarves.
If you carry on South past the Chaos Dwarves you will pass some Hill Giants which are killed for the Big Bones that they drop and come to a room full of Black Knights. You may want to kill Black Knights to get your White Knight Ranking up. On the East side of this room is a Jail Cell. To enter you must kill the guard outside, take the key he drops and use it with the door. Once inside you can talk to the Explorer who is locked inside and get a Dusty Key which is required for access to the deeper parts of the dungeon if you don't have 70 Agility or more.
Big & Creepy Monsters
If rather than going south to the Hill Giants you chose to go across the bridge and head North you will come to some Lesser Demons. On the West side of the area that the Demons are enclosed in is a gate. To pass through this you must use your Dusty Key on it.
Through the gate is a room full of Blue Dragons, Baby Blue Dragons and Blue Dragon Scales. It is recommended to be wearing an Anti-Dragonfire Shield in this room to protect from the breath from the mother Dragons, but if the room is busy then all of the dangerous Dragons will be getting killed by somebody else and you will be able to run straight through. The Dragon Scales are often collected because they are used in Herblore to make Weapon Poison and Anti-Firebreath Potions.
If you go all the way through the room and head South you will come to an area with 3 spawns of Baby Blue Dragons. These are killed because they give good Prayer experience when you bury the bones they drop. Next to these is the fishing spot used during the Heroes Quest to catch Lava Eels.
To the West of this is a passage that will lead you to a room packed with Black Demons. They may be a high level and able to hit a lot of damage, but there are places in the North-West corner of the room that you can hide behind and Range them from. This is also a good place to use a Cannon.
Heading out the North of this room to the East is the room where you will find one of the Scorpions during the Scorpion Catcher Quest. The area around here is also full of Poison Spiders that will poison you! There are two exits out the North of this room. The one to the East leads to some Hellhounds which are killed for the level 3 Clue Scrolls that they drop. The one to the West leads to the home of the Black Dragon. They are level 227, but can be Ranged from a spot to the West. West of the Black Dragons is the Fire Obelisk which is used to charge Orbs to make them Fire Orbs and up the ladder in the South of the area with the Black Dragons is the Water Obelisk. This does the same job as a Fire Obelisk, but makes Water Orbs, not Fire ones. From here you can also see the Catherby shore.
So there we have it. You have reached the end of the dungeon. You can either teleport out or walk back the way you came. Happy exploring!
Skeleton - level 22
When you walk into the cave you will see about 5 level 22 Skeletons. These skeletons can be killed by magic, melee or ranged. I recommend maging them as you can use Crumble Undead. Skeletons drops unidentified herbs, bronze armor and weapons, iron weapons and armor, half key, air runes, rune javelins, and coins. If you are killing skeletons for a slayer task it gives 29 experience.
Suit Of Armor - level 19
There are two suits of armor, both level 19 and give a good fight for a level 19! They will show up if you try to open the door to the Cauldron of Thunder to the east. These unfortunately don't drop anything and they don't give slayer experience. A good way to kill the Suit of Armor is by using fire bolt or a higher spell on it since it is made of metal and metal is weak against mage.
Skeleton - level 25
This skeleton is almost the same as the level 22 except it can drop from air runes to nature runes and iron arrows to steel arrows. Surprisingly when you get them for slayer task, you will want to go with the level 22 skeleton as this one only gives 15 experience on slayer.
Ghost - level 19
Ghosts drop absolutely nothing! The best way to kill one is to cast Crumble Undead on it, since it works well against undead enemies. They give 25 experience if you get them for a slayer task.
Giant Bat - level 27
Bats only drop bat bones and nothing else. The best way to kill him is range. They give 29 slayer experience if you have them for slayer task.
Chaos Druid - level 13
This guy drops Dragon Square Halves, various herbs, all kinds of runes, Chaos and Nature Talismans, Dragon and Rune Spears. They are best defeated using melee attacks while wearing dragonhides since he is mage base.
Black Knight - level 33
Magic Axe - level 42
Poisonous Scorpion - level 20
Chaos Dwarf - level 48
Jail Guard - level 48
Lesser Demon - level 82
Baby Blue Dragon - level 42
Blue Dragon - level 111
Black Demon - level 172
Poisonous Spider - level 64
Hellhounds - level 122
Monk Of Zamorak - level 45
Black Dragon - level 227