Anomalies

[Fallen Shaman]

When should I go and do the anomalies I discover? I assume that the possible research gain will be "equal" throughout the game? The anomalies that give extra resources to systems should be really good early on to get better expansions, right?

[Demon of Razgriz]

I usually wait to do anomalies after the initial expansion is over and I have my borders where I want them. In the early game, your science ships are more valuable with surveying so you can expand your borders.

[MtvYoloSwag]

I always do them immediately when discovered.
However, as demon said, it's definitely more ideal to only use science ships for surveying at the start of the game. Get your borders up where you want them, then at your leisure you can explore the anomalies you've discovered inside your borders.
While they're in your territory nobody else can survey or explore anomalies.
Also, at the start of the game, your scientists are level 1, so even an easy anomaly might take 100+ days while harder ones might take 500. Which is insane.