Note that if you just started you can likely "early exercise" all of your options right now by filing an 83B and paying the company the strike price. You generally have 90 days from the date of the option grant to do this.
The only benefit to exercising early is to avoid the AMT impact.
So for stock options, you don't start the clock on the long term capital gains rate or set a cost basis for your capital gain until you exercise the option and shell out the cash for the strike price.
There is all kinds of information on the internet about this if you don't believe me.
When you exercise ISOs, you are buying the stock at the strike price. It doesn't matter what the current market price of it is (other than for AMT). The cost basis is the strike price.
The "clock" for determine long-term vs short-term starts when you exercise, but it doesn't affect the cost basis. If you don't hold it long enough the gain may get taxed as short-term or ordinary income (usually no a difference except in a couple of states).