Q) Do I need an Arduino UNO or MEGA for this?
A) You do not need an additional board for this. The ESP8266 is THE board.

Q) I am able to upload the sketch, but when i run the sketch I see a lot of
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A) The power supply does not give a stable 3.3Volt. (Are you using power from USB port, that is most likely too less mAh, it is recommended to have 250mAh

Q) I am able to  upload the sketch, but when I run the sketch I see a lot of
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A) Have you connected your USB to TTL to ESP8266 like this? DTR to GPIO0 and RTS to RST. This solution is explained in Blink (without support of Serial Monitor). Just disconnect the DTR from GPIO0 to solve the problem. Or use 2 NPN transistors like explained in Blink using NPN transistors

Q) I am not able to upload the sketch. The message is “warning: espcomm_sync failed”  and  “error: espcomm_open failed”. Though I have normal results in Serial monitor
A) It could be anything. Check blink page . If that has not solved the problem check your power supply again. I added once a capacitor ( 220uF) next to the ESP8266 and that solved my problem.

Q) Where can I find the library for my Arduino UNO.
A) You do not need an UNO to use the ESP8266. The software is running on the ESP8266

Q) I cannot compile the standard examples.
A) That is correct. We need some help for this. Although the basic examples are already working.

Q) Where can I buy an ESP8266.
A) Where ever you would like. We bought them from aliexpress

Q) I got problems strange results when I using power from my USB port.
A) This is a common problem. Use an external powersupply