Electronic – RFID Reader Really Slow

It's possible, but most (if not all) commercially available tags just transmit their code upon reception of the carrier. The antenna is their only I/O.
What you need is a low-power microcontroller which is fed from the energy it receives from the reader's carrier. As a microcontroller it can do anything you like, including sending a code to the reader, and switching any external devices like LEDs. Of course these devices will need their own power source, since the power extracted from the carrier will be hardly enough to power the microcontroller. A good low-power microcontroller is TI's MSP430, "ultra low-power" in TI's marketing speak.
You'll need to have good antenna coupling to get a high enough voltage not only to operate the microcontroller, but especially to control the external logic. You may have to use level shifters which accept very low input voltages. For instance the Fairchild FXLP34 can work with input voltages down to 1V.

This gal made an RDIF tag with an AVR microcontroller and some clever tricks.

If you have the RFID readers connected to a network, you could do this pretty easily.

You need to separate your project into two parts - getting the rfid info into a processor/microprocessor of some sort, and then moving the data to another location. Both are pretty easily solved with off-the-shelf parts.

The problem I think is that you're asking/googlinh about transmitting "RFID Information" in particular. Once you scan a tag, you basically get data. Data is pretty generic. People spend a lot of time working on moving data around. Once the tag information is in the computer, it's just a string you need to send over some sort of medium, and if your medium supports data, it probably supports any actual data contents, RFID or not.

I'm not too sure how many of these things you want to make; an easy if not cheap solution would be to just slap an RFID reader on an Arduino, together with a GSM module (for cellular communication), or a WiFi module. However, this is a bring-your-own-code solution. If you're hoping for something off-the-shelf, short of using an full-on desktop/laptop computer for each node, I think you're out of luck.

If you already have a lot of RFID readers, there are plenty of serial/analog/whatever bus to wireless adapters out there, but they're not going to be labeled RFID-specific. Again, think data, not RFID.