Only natural fabrics get more absorbent with washing, so that'll mean your organic Flip inserts. They can take up to 10 washes (a good rinse will count as a wash) to become fully absorbent. Anything else (covers, pocket outers, microfibre inserts like the BG inserts and the stay dry inserts) you just need to wash once to make sure there are no chemicals from when they were made on them.
For the first wash, I do a normal nappy wash - so 1/4 a normal detergent dose and a good length wash (not a quick wash) at 30-60 degrees, then an extra rinse or two until there are no more detergent bubbles. You could use everything once they're dry. If you do that, just remember to change the organic inserts more often than you would normally until they've been worn and washed a couple more times (after a few washes, they should have had enough rinses to be pretty much fully absorbent). Or you could do extra rinses on the organic inserts before using for the first time. An alternative to machine rinsing is to leave them to soak overnight in a bucket/bath of water.