As I was going through Pioneers (and previously Emergence), I realized that there are some really effective, non-obvious strategies that can be really effective:

1. Piracy. Once you're in space, money is no longer an object: just take it. For very little risk and no harm to reputation, you can take the inventory of every capital ship's storage compartment just by shooting it. They're rich in gold, platinum, chromatic metal, etc. Staying close to the ship also makes it difficult for the sentinels to actually hit you even in an un-upgraded ship. As long as you don't attack the support ships or the independent storage ships, you don't lose reputation with factions.
2. Just spend. With essentially unlimited piracy money, just spend it to buy things. Don't craft if you don't have to.
3. Space Station Missions. At the space station, stack every single "kill" animal mission and "kill" sentinel mission. Opportunistically do them while doing everything else. Since they stack, you'll get significant payouts in nanites, credits and goodies for the same tasks. You can turn them in at any mission agent in any system. Chalk this up to the low effort required.
4. The Anomaly. (I actually think they should block the anomaly on expeditions, but…) You can use it for missions — raid depot missions every time — and to travel to otherwise blocked systems or to travel through the portal to other people's bases. It's another "easy" way to get access to things that would otherwise be inaccessible early.
5. Other Player Bases. Unlike the regular game, the expeditions are in really close quarters. At this point, lots of other players have already done some significant things in the relevant systems. For things like farming, skip the wait. Look for other player's bases at nearly every system. Someone has usually helpfully named a base "Farm". Keep an eye out for other things like that (AI Farm, Runaway Mold, etc.).
6. All fauna nanite reward. Recent updates made scanning creatures really valuable. If you happen on a planet with something like less than 5 or 6, it's totally worth it to find them all just for the nanite reward for finding them all — check the "discoveries" page and claim the extra reward. This is opportunistic, but low effort way to earn a bunch of nanites.

Also, for your own sanity:

1. Storm crystals. If you happen upon Storm Crystals, take them. Every one of them in reach. They're both easy money and (in both 1 & 4) necessary. It's worth the effort to get them.
2. Don't do things in order. Look at all the milestones. Lots can be done passively (e.g., walking around, earning cash, earning nanites, etc.). There are some you should do asap: like get the ship and multitool.
3. In expedition 1, the only time gate is Frigate Missions. You'll get a free capital ship pretty early. Take it. Then buy 3-5 really cheap Frigates and start those ASAP.
4. Sell upgrades for nanites. Since this isn't your permanent game, sell (and fast) upgrades you're not going to use for nanites. For example, exocraft and underwater upgrades.
5. Get the S-class ship, early. Unless you stumble across something better early, get the free S-class ship. It takes only 3 space anomalies. Any you get free anomaly detectors from warping to the red, blue and green systems. You can do that via the Anomaly without having to upgrade your ship. Save your expansion slots and upgrades for this one.
6. Abandoned System. Since this gets asked: install an economy and conflict scanner. Look for a [edit: blue (not red)] or green system with a Race and Economy but no conflict data. This is an abandoned system. I suggest you do this immediately after going through a black hole. Note you won't get the milestone if you find an abandoned system someone else already found. So the further away from the expedition you go, the better. [edit: thanks to u/f4780y, you can get away with just an economy scanner. Look for a 1⭐ system with race but no economy data].

I hade a bit of luck. But here are two Expedition 1 Exotics for those who read this far:

https://preview.redd.it/v7dvwn6k6z281.png?width=1900&format=png&auto=webp&s=ea94333b2df1538b00ced6618899d6b0cc2b6948

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https://preview.redd.it/kdg0103n6z281.png?width=1875&format=png&auto=webp&s=b23eb2e4df947f91c7db4ea7a818da98f0eb5063