The MBED microcontroller modules are merely the respective NXP microcontrollers implemented in a PCB with a wide DIP form factor, with supporting components.
In other words, they are like an Arduino Nano or Pro Mini. They are not made by NXP just like the Arduinos are not made by Atmel.
There is nothing unique about the mBed that would significantly alter the on-board microcontroller's ratings.
Thus, the input and output voltage and current ratings would be the same as for the respective microcontroller used in the board in question. The relevant datasheet will provide the information you need.
From the question, a microcontroller will typically not be able to source or sink 300 mA, that the relay drive coils require. Not only that, sourcing that much current from the regulated lines on the mBed boards will either overheat or destroy the on-board voltage regulator on those boards.
A recommended approach would be to power the relay's primary from an external power source, switching it via a logic-level MOSFET or BJT: The 2n7000 might be borderline suitable, as it is rated for 400 mA not counting requisite derating.
The MOSFET's gate would then be driven by logic outputs from the mBed, suitably protected by a small resistor (100 Ohms should do) in series with the gate. Also, a 10 kOhm pulldown resistor on the MOSFET gate would address spurious switching of the FET when the logic outputs are in high-impedance mode.
Best Answer
To expand on my answer in the comment above. Take your mbed out of the packaging and insert it into its' new semi-permanent home on your breadboard. Turn on your computer and connect your mbed using the USB cable which came in the box (or any other similar cable). As Rob says it'll show up as a flash drive called MBED, I've not used one on a Windows computer but I don't think you have to install any drivers (on Linux it's all ready to go out of the box).
On the mbed drive is a single html file, open this using your browser of choice and it'll take you to the mbed.org site and ask you to register (you have to register to gain access to the online compiler). The good thing about this process is it automagically attaches your mbed serial number to your registered user (you can't register without having a valid serial number at the moment).
The getting started instructions on the mbed.org site (and on the leaflet that came in the box) will walk you through downloading the compiled binary example to make LED1 flash on and off. Once you've downloaded it, copy it to the mbed drive, and press the reset button on the mbed (it's in the middle of the board) you should now have a fully functioning blinking LED.
Next have a look at the program in the online compiler and change it. Once you've got it going then you can start hooking it up to various external things. There are various instructions in the Cookbook and Notebook pages at mbed.org (including how to hook it up to a dual digit 7 segment display by me).