If you leave a company with exercised shares before it goes public, how do you get access to these shares after IPO?
When the shares were exercised (meaning you paid the strike price the option to purchase shares had been set at) you should have received a stock certificate for the shares you purchased. This is your proof of your "piece" of the company. Once a company goes public those shares still exist (maybe in slightly different numbers based on the offering share count) and you will be able to trade them in the public markets. The key is to make sure you have a certificate that demonstrates the number of shares you purchased. If you don't have that contact whomever manages this function inside the company, usually in house legal,
I agree with Brenda with one small addition. Some times the company will keep custody of the shares for ease of administration. If you do not feel comfortable with the company having custody, you should be able to request they send the certificate to you.
Here is another small tidbit about holding pre-public shares - they will not be registered and fully-tradeable upon going public. They will get registration through Rule 144 and I would expect the underwriters of the company going public will require all shareholders to sign a lock-up agreement limiting your ability to sell shares for 180 days after the IPO.