How can I see deep underground?

There are two primary methods I use.

1) Craft a bunch of torches and drop them along the way. They have a fairly large light radius but you will still need a hefty supply of them if you're doing any extended mining.

2) Craft the Lantern Stick. You wear this on your back and while it gives off a much smaller light radius than torches, it has the benefit of letting you just dig away like mad without ever having to switch over to torches and drop them. Particularly useful when you're just digging a hole straight down. Also operates while underwater, unlike torches. However, I don't think this recipe appears until after you craft the first star map upgrade, so you'll have to get by with torches until then.

Here are all the ways you can light up a cave as of 1.0 stable, loosely ordered by usefulness and difficulty in obtaining

1. Flashlight / Matter Manipulator scanning mode - the Matter Manipulator's light cone is actually slightly larger than the flashlight's, but the flashlight can be welded with another weapon or tool, like the rope, which is useful for descending down steep caves. A flashlight can be found in the ship's locker at the start of the game.

2. Torches - craft 4 for 1 x coal + 1 x log using the crafting menu (default: C). Useful for seeing around corners, since you can place them anywhere there is a background wall within range of your Matter Manipulator. Doesn't function underwater.

3. Flare - single use (lasts 20 seconds). Found in chests, pots and capsules, or crafted 2 for 2 x Plant Fibre, 1 x Volatile Powder (from Core Fragments), 1 x String (from Cotton) at a Foraging Table. Useful for seeing around corners since this is a thrown item.

4. Glow - The Glow status is given temporary by the Yellow stimpack (crafted from Apothecary using 1 x Medical Syringe, 1 x Yellow Petal, 4 x Plant Fibre) or when any part of the Glow armor set is equipped in armor slot (not cosmetic). This somewhat limits their usefulness.

5. Backpack - A Lantern Stick, Halogen Pack or Xenon Pack will provide light if equipped on the back slot (not cosmetic). This means that they can only be used on planets with no hazards.

6. Light EPP Augmentation - the Light I, II and III augmentation can be used to give your EPP the same lighting capability as the backpacks, at the cost of the augmentation slot. EPP augmentations can be found rarely in chests and other containers when exploring.

7. Pet - some monsters and critters emit light. Capturing one with a Pokeball Capture Pod will give you a mobile light source that has a mind of its own.

8. Weapon fire - weapons that causes status effect (energy, flaming, poisonous, frozen) also emit a small amount of light when activated. It's not a lot, but it is enough to light up nearby enemies, and in the case of projectile weapons, show you where the walls are.