The MFRC-522 you linked looks good. You will need to do a small amount of soldering in order to attach wires to the MFRC-522. BUT you can also get it to work by manually holding the wires to the metal contacts on the MFRC-522, which isn't ideal but it can work in a pinch.
If you don't have any soldering equipment or experience, you can find mfrc-522's on amazon that have been pre-soldered. This one for example comes with the leads already attached. You might need to do some browsing to find one available in your area.
Pairing a pre-soldered MFRC-522 and some male to female wires, that usually come with arudino kits, should make everything plug and play.
You might need to go to arduino's website to get the sketches that go with the MFRC-522. Specifically the "dumpinfo" sketch.
That guide that I linked initially should get you on your way.
I had never used an arduino before I started playing with this, so i think this might be helpful info for newbies. After you have you arduino and rfid all hooked up, and you have the "sketch" uploaded and running on the arduino, you need to go to Tools > Serial Monitor.
This pulls up a window that will be displaying the UID for the cards/tags. And it'll also be the window that will tell you when it's ready to write to a card/tag and when the writing is done.
edit:
If anyone reading this is struggling with connecting their arduino to the rfid. This info is also in the header of the sketches that correspond with the rfid.
9 -> RST
10 -> SDA
11 -> MOSI
12 -> MISO
13 -> SCK
GND -> GND
3.3V -> 3.3V
This is the board
This is the NFC reader/writer