Nowadays lots of people use the word "google" as a verb, past tense "googled", which generally means to search the web using Google's search engine. When used as a gerund the word is typically "googling". (I'm not sure if these words should be capitalized, but that's not relevant to my question.)
Is there a common word or phrase that can be used in the same way, but that doesn't imply using any particular search engine? I'm curious because not everyone I talk to uses Google.
Example sentence, as a verb:
I _______ for the words "cute kitty" and found lots of results.
Example sentence, as a gerund:
I did some _______ but couldn't find anything relevant.
The best I can think of is "web search" or "web searching" but it strikes me as awkward. I'm hoping there's something a bit better and more natural sounding, especially if it's a single word.