To save an URL into a different file name the "-O" option is the best way to go. If you have special characters like ? or & in your URL enclose the URL in double quotes.
This is how it should look:
wget -O testme.gif "http://larshaendler.com?test=me"
SSL source without checking the certificate If you have a ssl source and do not want to have the certificate checked:
wget --no-check-certificate -O testme.gif "https://larshaendler.com?test=me"