How do I figure the cost basis of stock that has split, giving me more of the same stock, so I can figure my capital gain (or loss) on the sale of the stock?

When the old stock and the new stock are identical the basis of the old shares must be allocated to the old and new shares. Thus, you generally divide the adjusted basis of the old stock by the number of shares of old and new stock. The result is your new basis per share of stock. If the old shares were purchased in separate lots for differing amounts of money, the adjusted basis of the old stock must be allocated between the old and new stock on a lot by lot basis.

Do I need to pay taxes on that portion of stock I gained as a result of a split?

No, you generally do not need to pay tax on the additional shares of stock you received due to the stock split. You will need to adjust your per share cost of the stock. Your overall cost basis has not changed, but your per share cost has changed.

You will have to pay taxes if you have gain when you sell the stock. Gain is the amount of the proceeds from the sale, minus sales commissions, that exceeds the adjusted basis of the stock sold.

How do I calculate the average cost method of a mutual fund if the fund price splits?

If your mutual fund splits, or adjusts its price, it is treated like a stock split. Your total basis doesn't change after the split, but since you now own more shares without paying any more money, your per-share basis will decrease. To calculate your per-share basis, divide the total cost that you have invested in the fund (minus any shares previously sold) by the current number of shares that you hold.