1. Cellphones also have many functions like going on the Internet.

2. Cellphones also have many functions like logging on the Internet.

3. Cellphones also have many functions like logging in on the Internet.

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Are they all the same in meaning?
What is the meaning of 'going on' in Sentence One? Would you paraphrase the expression? Thank you.

They all have the same meaning. "logging on (or in)" is more correct than "going on". "going on" is more slang. And, in sentence 3, "in" and "on" are redundant, in that they mean the same thing. So sentence 1 is the best.

To paraphrase: "Cellphones have many functions, such as logging on the internet."