Veteran Civilization players might be used to going after Resource Pods because they feel like abandoned villages. You step on them and get a reward. But in this game, expeditions are the real fruit of the land. The rewards from these sites are way more than anything you're ever going to get from the Resource Pods (which primarily have Culture, Science, Energy, or a minor orbital if you're extremely lucky). Expedition sites have the potential for major Research progress, extra population, affinity Experience, quest rewards, free units, and more.
This is big stuff.
You should pursue expeditions as assertively as possible throughout the game, even taking them from territory near your neighbors if you're willing to take a minor diplomatic hit. Heck, if there is only a single site in that area, take it, and then promise to stop doing expeditions near that neighbor. Doing so actually helps your diplomatic ties with them in the long run, if you stick to your promise!
Early on, have your explorers run through the world, grabbing Resource Pods and looking for future expedition sites. Sites that you uncover outside of miasma should be grabbed immediately. Return to your city afterward to stock back up on the supplies your explorers need for additional expeditions.
Resources inside miasma are tougher. They often take 10 turns to uncover, and your explorer might die from the miasma before that happens. To counter this, look for upgrades that hasten improvement times. Helping Hands (a Prosperity virtue) is great for this. It allows explorers to get their expeditions done inside miasma without dying. They'll take tons of damage, but it's worthwhile.
Every time a station is destroyed, an orbital falls, or an alien nest is cleared, you get additional expedition sites. Scout for these throughout the game, and don't retire your explorers. Even when there isn't anything clear to go after, have them stay inside your borders and Fortify. Wait for a glimpse of a site, and wake them up to return to duty.
As you get deeper into the game and can spare some military forces, create a Rover to accompany your explorer. Rovers can actually keep up with explorers or pass them on open ground. That makes them great escorts, with enough firepower to hold off small attacks by aliens or other nations.
In aquatic maps, you should consider quest or virtue upgrades that give explorers more expeditions per trip. This is only decent when you have roads and easy access to everything. Once you must cross oceans just to explore, it becomes fairly costly to return to a city. Having more expedition modules can mean the difference between getting one expedition or several!