Can I use arduino's 3.3 V output directly to esp8266?
You should search for this data yourself, but since you are saying you are a newbie I'll tell you how to find this.

First of all you should search for a document called "datasheet". On a datasheet the producer writes all the relevant data about his product.

In your case, you should search for the electrical characteristics of the ESP8266. For instance here you have one of the datasheets. On page 13, there is the "Power consumption" chart, which says that the worst case has a typical current of 170mA. I suggest you to raise it a bit; let's say you need 250mA for the ESP.

Now, the arduino. You should search for the schematic of the board or read the part number of the 3.3V regulator. Since it's easier from the schematic and they released it on the web, you can see that the part number of the 3.3V regulator is LP2985. Searching for it on google will lead you to the texas instruments webpage, with the datasheet. Here you can see that the maximum output current is 150mA, so below the required current. Answer: you can't with arduino.

As for the FT232 cable, you should search for it because there are different models. Search on google and pick the first link pointing to the ftdi website.

As an example, the TTL-232RG has this datasheet. On page 9 they say that the TTL-232RG-VSW3V3-WE has an output current of 50mA, so you can't use it. The TTL-232RG-VREG3V3-WE, on the other side, has a maximum output current of 250mA, so you can use it.

If you have further questions, just ask here ;)