Your first error implies your script uses outdated code. eregi() is a function that cant be used with PHP higher that 5.2 which your new hosting probably has (5.3 or 5.4 is most common now). Since youre not a programmer easiest solution would be to find an updated version of your formmail script.
You can always search for alternate solutions, see this Good alternative to eregi() in PHP if you still want to try to debug the code yourself