It depends... What I suggest that you should do is to look at the documentation of each of the methods for which you get the deprecated method warning and see what it recommends (if anything) and then look at your use of the method and see how easy it would be to change. For example, some methods have been superceded by newer ones which would only take a minor syntactic change. Others might require significant rewrites to your program to address.
Also note that you have to keep in mind whether or not your program needs to be run under an older version of Java. I.e., if you've got an applet built to work under Java v1.0.2 and you want to keep that capability then you shouldn't change anything that would break that.