Actually there are two ways to get Exp. Shares, although the one you get from Rowan's assistant is definitely the quickest. Info on both methods below are sourced from this wiki page on Guide Strats:
From Rowan's assistant gives you an Exp. Share after you've seen at least 35 Pokémon registered in the 'dex (seen, not required to be caught). This happens in the gate connector in southern Eterna City that leads onto Route 206/Cycling Road.
Multiple Exp. Shares can be acquired as repeatable prizes from the Pokémon Lottery in Jubilife City. You'll get an Exp. Share every time you have a Pokémon who's last 3 digits of their OT ID match the last 3 digits of the daily lottery number.
You must atleast seeing 35 or more Pokemon to get it Go to Eterna City, then go south. Talk to the scientist guy (Professor rowan's assistant) right infront of the door He will give you EXP share if you have seen 35 or more Pokemon
