When you upgrade from an old version, you may need to jump through a number of different sets of dependencies. I am afraid it is inevitable.
I also suggest doing it in stages, and doing a bit of testing in-between version jumps.
It may be easiest to backup your code and database, transfer it to a development environment such as your own PC (where you should be able to install/configure any version of PHP/MySQL), deal with the upgrades to the latest version, then reupload it to the server (running latest PHP/MySQL). And if you stuff up, you can just return to step 1 and try again, without damaging your original install.