Tips for running 1st Place

General Tips
Longer trek legs are better than short for conserving sanity and time.

Always follow the compass directly to go fastest because there are no magnetic mountains in Tourist difficulty.

Load up on sanity items early when shopping and collect loot from easy-to-access locations along the way if youre not already stacked with supplies.

It is faster to attack enemies and then immediately flee before rolling than it is to let the enemy attack you first and then flee.

The only quests worth doing are all the Expedition 1 quests, as well as '80 days' and 'Find an Instrument' for the later expeditions. Doing others will waste time to complete and failing them hurts your fame which is hugely important.

Dynamite can save you a lot of time or get you out of a pinch if you take a bad turn into mountains.

Saving a small amount of funds after shopping (20-50) is a good idea in case you find the pyramid "too fast" (ie not enough loot) in the following Expedition.

Buy all the weather balloons and use them on Expedition 5 to expediate it, being the longest segment in the run. Somewhat related, you also always want to use stone circles: In E2-E4 I use them to reveal shrines and use the info to plan out the rest getting the most loot in the least time. In E5, if I have weather balloons, I will ideally find the stone circle in the first 30 seconds, then reveal caves (bc there are usually several caves) and then immediately use any balloons I have collected. If you get 10-15 balloons this will pretty much reveal all locations, greatly reducing the time and risk needed to find the seal and pyramid.

And finally: Never go to the desert.
(why?): Crossing desert hexes take a relatively long time, refilling water wastes storage and small amounts of time, and you may have to fight mummies to get loot in tombs. I never fight.

Expedition 1 guide
Always take the first quest: It reveals the village, gets you money sometimes, and gets you the standing needed to recruit an animal. It is worth it to trade for gems and any other loot here as well. I also like to get the horn instruments if they are available as well. Never visit a shrine before going to this village, even if it means backtracking: you will either miss out on getting an animal or you will waste time and resources trading to gain the standing to do so. In Expedition 1 make sure to get both shrines. To get the best time possible you need a lot of fame early and the E1 map is smallest. Plus if you take too much time its a quick reset.

Finishing the Run:
Do not play Expedition 6, sail home immediately upon arrival. I believe it is impossible to beat my time if you play Expedition 6.
I have gotten 1st Place with fame as low as 1870 but I have also gotten 3rd place with fame over 2200. Usually I only really feel comfortable that I'll get 1st if my fame at the end of E5 is well over 3000, but there is an generally negative relationship between loot collected for fame and time elapsed, with some variability based on shrine and pyramid location across the expeditions.
So that means to get the absolute optimal best possible time you would really want to live on the edge of necessary fame and hope for a great time paired with a poor showing from the NPCs, the latter of which appears to be completely out of the player's control. Do with that what you will!

2024 Update: The strat now is to skip both expedition 5 and 6 and hope for the damn best

DoubleTDoubleCT

Curious Expedition

Spikeawe

Deetec

Any Difficulty

Expedition Difficulty

Game difficulty (either any difficulty, or expedition difficulty)

Difficulty

Medium AKA 'Curious Expedition' Difficulty

BerryTime