How can I navigate and mine hell safely?

There are several problems with hell and several ways to overcome them.

Hellstone

• Wearing an Obsidian skull will stop you being bunt from standing on it.
• Putting sand on top of the block before mining will remove the lava that is created.
• The buildings in hell give an excellent vantage point for mining the hellstone through the wall.

Lava

• Rocket boots allow you to cross large lava lakes easily, they are sold by the npc Goblin Tinkerer.
• Grappling hooks or Ivy whips work in similar ways.
• Placing sand above the lava will remove it when the sand falls into it, but you have to build up from the bottom.
• Water walking potions also allow you to walk on lava without being hurt.
• There is a trick with flippers that allow you to swim above the lava.

I've had good success mining with something (a glitch?) a friend and I found:

Hellstone
If you mine to the bottom of hell, so your player is standing on the bottom of the screen, you can throw Dynamite or bombs on the ground and the blocks off-screen will break. It seems like it keeps track of at least some of the blocks off-screen. You can then fill that area with lava and expand it with bombs or dynamite. Note: The area CAN fill up, the lava doesn't disappear so make sure you make it deep before you start mining. :)

Then, we usually just mine out the ash between the bottom of the hellstone and the bottom of the map and let the lava fall off-screen while mining beside it. We also mine from bottom to top.

Tunneling
Because of the way we mine, we usually start by making a tunnel straight down through the ash and avoiding hellstone. Then, it just expands across the bottom based on how much hellstone we mine and how much we have to expand the off-screen "pool".

Crossing Chasms
I usually just use the extra ash I've collected to make a 1 square "bridge" across.