You are taking an hard way. You’d need at least an RS232-to-TTL translator (or an USB2SER kit, or the great BusPirate v3) since that reader needs bidirectional comm (and if you want to do something more complex it’s not easy – at least I couldn’t make it detect multiple cards…).
For an easier start I’d recommend an ID12 that already automatically drives a buzzer when it recognizes ANY 125KHz card (doesn’t check the code, just card presence). If you want, you can then hook up a serial receiver to make a PC (or an Arduino) receive and decode the card serial number.
Once you (or your kid) have enough experience, you can start studying more evolved tags like the 13.56MHz ones (that allow read and write).
The RDM880 reader offers two outputs for LEDs an another for a buzzer (too bad wiring diagrams are missing from the docs: do LEDs need the series resistor? Does it output 5V when active or shorts to ground? [well, a multimeter is enough for this, but it would be better to have “official” docs about it!] …).
A big difference between 125KHz readers and 13.56MHz ones is that the former uses asynchronous signalling (they output a “message” as soon as a tag enters the field) while the latter requires polling (you have to interrogate it to receive a card code).