The fee is optional, but without it miners have no incentive to include your transaction in the blockchain. They may just abandon it and leave you with a hassle. You should pay the tiny fee, it is still the cheapest way by far to send money.
As far as investing... That is tricky. "Investing" in BTC is extremely risky and you should be prepared to loose everything. Bitcoin is not designed as an investment, it is a payment system and I would recommend thinking of it that way. But if you want to do it anyway for fun or whatever. Then start with 1 bitcoin and try selling it high, then buying it back low. Make a game of it!
Remember to also track what you put in and the value when you sell. You will be required to report any profits you make and pay capitol gains.
Good luck fellow peer!