Is your 1918 penny rare?
It depends on what you consider rare to be, but yes, there are certainly fewer 1918 pennies in America than more recent dates.
Please see the Related Question for values.
The 1860 bronze (not copper) Penny of which there are 16 variants, only one of which is considered to be extremely rare. Wreath of 16 leaves, all other have 15 or 17 leaves. The 1863 bronze Penny with a slender 3 is considered to be extremely rare. The 1863/1 bronze Penny (3 over 1) is considered to be extremely rare.
For coins in similar condition, the 1918 Australian Penny would be worth much more than the 1917 Australian Penny. An 1859 Australian Penny was never minted. The first Australian Penny was issued in 1911. With the possible exception of traders tokens (with the traders business name on them), there were no "Australian" pennies minted prior to 1911. The only Pennies circulating in Australia prior to 1911 were British Pennies.