I've exercised before. Typically, you email hr and say, "I want to exercise"; they send you some paperwork which you fill out; you write the company a check. DO NOT DO THIS BEFORE UNDERSTANDING TAX CONSEQUENCES. You will typically pay tax on the spread between strike (your price per option) and the fair market value (fmv) which is set by the board and often updated quarterly. This can also be a backdoor way of a board tightening those golden handcuffs; if you where early enough the taxes may well exceed the strike price. You should also be able to get the fmv by asking. Keep in mind these shares you're buying may well be completely illiquid and the irs wants their taxes right now anyway.

A numerical example: 20k shares with a strike of $0.11; fmv of $0.39. Then I write the company a check for 2e4 x 0.11 = $2200 dollars and report income to the irs of 2e4 x (0.39-0.11) = $5600 (for amt).

A nuance is if the company is succeeding, it can be worth it to buy options when they vest; it starts the clock ticking on long term capital gains and can roughly half your tax bill if and when you can actually sell the share. Which reminds me: you will pay taxes twice: once when you exercise the option to turn into a share, and again when you sell the share. If you are lucky enough to go public the company will often get a firm that handles all this for you and just gives you a check net of all taxes.

A good accountant will cost $500-ish (or less) to go over your situation in detail. It's worth the money. If you already pay ab accountant, not someone at hr block or similar people who just know how to fill out paperwork, they may go over your situation for much less money.

Also, you must understand amt; that can bite hard. If you don't understand amt, see that accountant.