I know has been out online but it is worth posting so others can see it. With the changes to engineering, completely maxing out a weapon or module can take 5-7 of a set of engineering items to complete. Data and manufactured items can be had through travels and missions, but elements can take surface prospecting.
One way to make this quick is to have the Thargoids do all the work for you and farm the small drop objects from barnacle sites. Barnacle elemental drops are the same as the composition of their host planet. If a planet has 2% polonium then 2 Thargoid fruits out of 100 will be polonium. The trick is finding a still living barnacle on a planet that has the elements you want. That is a reason why a post like this is nice, so others can post systems and planets for barnacles and elements that are popular for upgrades.
Another aspect to keep in mind is that there seems to be a static number of Thargoid Pickup Fruit no matter what size the barnacle site is. Visiting a barnacle forest will have many empty branches and the pickups will be spread out. A small barnacle has many pickups per branch. I know I have seen at most four pickups on a single Thargoid Barnacle spine. Also board flipping (If you do, great. If you don't, I won't judge how you want to enjoy your time) resets all the pickups.
If you have any sites to share, please do so. A detailed surface scanner is needed to find out the planetary composition and a SRV is also necessary. Also watch your fuel. The ship still uses it when landing and idle.