In future please focus on asking one question per thread rather than multiple :)
1) Where do you buy Bitcoins (USD/SGD/PHP/OtherCurrencies to Bitcoin)?
A number of exchanges allow you to deposit USD or other currencies and purchase BTC, usually after some verification process. You can search for exchanges or have a look at other questions with the buy-bitcoins tag here.
2) How do you transfer it to another wallet (i.e to Poloniex)?
Your wallet will generate an address for you, which will usually look like a random numbers and letters around 30 characters long, starting with a
1. You can enter this address into the withdrawal section of the exchange you chose, and they will send the BTC to it.
3) When transferring an amount to Poloniex, does it have a low transaction fee?
No, transaction fees are the same no matter where you are sending from / to. At the time of writing, I suggest a transaction fee of around 100 sat/byte, which would give an overall fee of roughly $5 per transaction. If you only have a small amount of bitcoin, you can keep it on the exchange without withdrawing to a wallet if you want to avoid this fee.