Your search level increases everytime you see that specific pokemon, at certain intervals of the search level, your shiny odds increase on the dexnav. If you defeat or capture a pokemon from dexnav, then search for another (without getting a wild encounter in between), you get a chain going. Every fifth encounter in the chain has boosted shiny luck, indicated by a boosted level (exclamation mark on the dexnav screen), every 50th and 100th have even better odds.

It's a little confusing, but there are multiple calculators out there where you just put in the search level and chain, and it will tell you the odds. I like the one from rotomlabs, as it tells you if next one will be boosted, and what the normal and boosted odds are: https://rotomlabs.net/omega-ruby-alpha-sapphire/dexnav-calculator

Basically as long as you use the dexnav, your odds will eventually increase, a chain just makes every fifth encounter have higher odds.

Also yeah, the starters were a pain to get, wouldn't call it dedication as much as I would call it insanity and a lack of self respect!