Hello,
You may say: I logged on, I logged on the Internet, I signed on, I signed on the Internet, I connected to the Internet, I dialed up (if you use a modem hehe) to connect to the Internet. I am sure there are other ways to say this, but these are the first that come to my mind. They are all very common and I would recommend using one of them instead of the "I got on the net," which sounds clear, but not very elegant. ;-)
Best,
Paperclip013