While the solution is straightforward, here's some hints to keep you moving towards your goal. Most of the following hints will not terribly apply if you're only looking for this achievement. This was done with Master Prospector in mind.General Hints
*** KNOW WHEN THE GAME SAVES!
The game only saves when you return to Tumbleton or buy something from the shop. If you're having a bad run, pause, select Exit to Menu, and try again. If you'd like to undo a death, dashboard when you die and exit the game completely. This will reset you to your last position. When buying from shops, browse first, figure out what you want to buy, then exit to menu, restart, and immediately run and get the item.
** Don't talk to anyone you don't have to. The next quest is typically some commentary before the new goal of "go back to what you were doing." When you do end up talking to someone, like when a shopkeeper comments on your new gear, mash
** Don't buy any further pickaxe upgrades or lantern upgrades. It seems counter-intuitive at first, but you'll be using the drill first and foremost to dig, and you typically shouldn't be in the mines for more than three minutes at a time.
** The best upgrades to get in order are Pouch, Drill, Steam Punch, Water, Health. The exception is the Pressurizer, you likely won't be using Steam Jump enough to notice.
** Always have a teleporter in your gear. Since you won't be buying pickaxe upgrade, this frees up enough orbs for 12 teleporters. Once your inventory is filled and you can't pick up anything, drop a teleporter, drop the money, get any upgrades, buy another teleporter, and get right back into it.Speed Run Hints
** When you first arrive in Archaea, don't leave until you have $8 in Trashium and Copper. Once you have that, you can buy the first upgrade for the pickaxe without talking to everyone.
** Beeline for the Old World. You'll still need to get Run and Steam Jump on the way to the drill, but you effectively want to go straight down as far as you can with little deviation. The money obtained in Archaea simply isn't worth the time. If you end up going by the optional caves, take them for the orbs.
** The Old World can be taken fast at first, but the money starts getting good around the Steam Punch cave. Even then, the priority is still getting to Vectron as quick as possible.
** Depending on your skill level, you may want to skip the sequence breaks. You'll need to come back with the proper upgrades anyway, so you'll likely save a few minutes, but if you can consistently make the dynamite jumps or edge jumps, or have the patience to keep at them until you do get them, it not only saves time but gives you a nice chunk of change for upgrades early on.
** In Vectron, prioritize getting the Mineral Detector first, then start again from the top of Vectron and clean the place out. This will significantly help getting the later upgrades.
** If you didn't do the sequence break earlier, once you get Static Jump, go back to Archaea Cave I and use the double jump to get higher on the map and into a generator room. Don't forget to empty out the secrets here! Same goes with Cave III, although that cave is much larger. Consider cleaning out some of the area with the Mineral Detector instead of just teleporting directly back once done.
** For the final boss, it's best to have the Steam Punch maxed with Hydraulic Fluid to speed up the charge time. Maxed health and Damage Reduction aren't critical here. Destroy the generator, get on Voltbot's head and hit three fully charged punches to start the next round.
For reference, here are my relative times. I'll include both the time I could have beaten the final boss as well as when I actually did to delineate between a regular speed run and four stars.
Archaea finished: 0:19
Old World finished: 0:50
Vectron cleaned out: 1:33
All generators destroyed: 1:41
(Final Boss available)
$20k and 150 orbs: 2:09
Final Time: 2:13