To upgrade all of your ship modules to level 1 you will need a total of 420 common samples and 30 rare samples.
Common samples spawn on every difficulty, and the easiest way to get them is on the "Eradicate" missions. This is because around 10-15 of them spawn in a very small area, and after you eradicate everything, you can go around and collect them all in about 1-2 minutes.
Rare samples only spawn from the Challenging difficulty (Number 4) and onwards. They also do not spawn on the Eradicate missions, so you will need to play a mission with a bigger map (e.g. a 40-minute mission).
The best way to find all of the samples on the map is to find the radar dish sub-objective, which reveals everything on the map including where samples are located and will be marked on the map by a jewel looking symbol. However, there is a chance you will not get a radar dish objective so don't always expect it. Samples will also spawn at main and sub objectives as well as nests and factories, so as you do the mission itself be on the lookout.