Is there a more general term for "googling" that doesn't imply a particular search engine?

Actually to google is becoming a general term to refer to any search engine:

verb (used without object)

• (often lowercase) to use a search engine such as Google to find information, a website address, etc., on the Internet.

(Dictionary.com)

To google:

• As a result of the increasing popularity and dominance of the Google search engine,1 usage of the transitive verb to google (also spelled Google) grew ubiquitously. The neologism commonly refers to searching for information on the World Wide Web, regardless of which search engine is used. The American Dialect Society chose it as the "most useful word of 2002." It was added to the Oxford English Dictionary on June 15, 2006, and to the eleventh edition of the Merriam-Webster Collegiate Dictionary in July 2006.

(Wikipedia)

As in:

What did your search reveal?

On Wikipedia there is a whole subsection of SEARCHING devoted to Computing Technology.

Computing technology

• Search algorithm, including keyword search
• Search engine technology, software for finding information
  • Web search engine, a service for finding information on the World Wide Web
  • Enterprise search, software or services for finding information within organizations
• Search and optimization for problem solving in Artificial Intelligence

I appears when using search in the proper context it will be known that you used a search 'engine' to peruse the internet for information.

In more formal contexts, one can perform an internet/online search/query without ever mentioning Google:

We performed an online search to identify relevant conference proceedings, journal articles, reports, and academic theses. — Andreas Freitag, Applying Business Capabilities in a Corporate Buyer M&A Process, 2014.

The Rugge Group performed an online search of the trade press to identify leading software developers and to find articles on the cost of developing software. — Ronald G. Albright, Electronic Communications for the Home and Office, 2018.

You have now performed an online query for “web design company” and now the results page is sitting there just waiting for your instructions. — Denver Online Pro, 28.10.2017.

In an attempt to locate the employer, a WHD investigator performed an internet search and located a business going by a similar name and listed as a farm instead of being listed as a trucking company as the complaint indicated. — Gregory D. Kutz, Department of Labor: Case Studies…, 2009.

In a more conversational tone, where performed would seem out of place, one can simply do the search:

Maybe you experienced some of these feelings the last time you did an online search. You had a question, launched an Internet search, roamed around the search results, checked out some of the more relevant items, and hopefully got what you needed. — Patricia A. McLagan, Unstoppable You, 2017.

“I did a search online and found a couple of sites that are really good. One of them was exactly what I was looking for, it was all about the topic I was working on, another site had a load of quotations from other people that were really useful. — James Carmichael, Challenges in Counselling: Research, 2013.

As a sidenote, the German verb googeln (ich google, du googelst, er googelt, ppart. gegoogelt) first made its way into a dictionary in 2004 and is required vocabulary for the B1 certificate in German as a foreign language from the Goethe Institute.

In the days of the web before Google existed (mid to late '90s), we just called it "searching" or "performing a web search" (web search would be the common noun). Doing so usually entailed using 3 or 4 different search engines to find what you were looking for, as no particular search engine was very good, and each would give different results. Some I remember using include Yahoo!, Alta Vista, Web Crawler, Excite, Lycos, and Ask Jeeves. None were overwhelmingly dominant.

Google was superior enough to the competition at the time it came out that most people abandoned other search engines. It could usually get you what you wanted on the first try. That's when "googling" became the generic verb for searching the web.

Perhaps browse, browsing the web.

I used to use look up

to try to find a particular piece of information by looking in a book or on a list, or by using a computer
_Macmillian

And, this is the phrase taught on Sesame Street (during Elmo's World when Elmo is consulting with Smartie the smartphone).

In your examples:

I looked up "cute kitty" and found lots of results.

and as a gerund (up is dropped)

I did some looking but couldn't find anything relevant.

Scan(ed)?

"I scanned the web/net last night looking for an answer"

If I'm doing extensive googling for something at work, I usually refer to it as researching. As in:

I researched cute kittens online and several sites agree that torbies are the cutest.

Research suggests that I compared and compiled the results from one or more web searches, rather than did one search and opened the first result that seemed like a good match. So it isn't a perfect substitute for "Googled" but it may be a better fit for when you want to convey greater depth than just a casual one-off search. See also: wikihole.

I searched online for the words "cute kitty" and found lots of results.

I did some searching online, but couldn't find anything relevant.

"Web search" beats "online search" and is competitive with "google" on Google Ngram viewer

The real meaning of "to google" is to do a cursory search of the publicly accessible Web. This has the connotation that the search is far from exhaustive. A lot of quality online information requires registering and even paying. E.g., when you "research" a job candidate, you'll probably begin by logging into LinkedIn. You may even pay for an online background check. Contrast this with the quip "let me google that for you," which suggests that a question is so easy to answer that googling it is all that is necessary.

"Web search" arguably better captures this meaning of "google" than "to research" or to "search online."

Q: Is there a more general term for “googling” that doesn't imply a particular search engine?

I sympathize with your plight as I have grown weary of "googling" [it] as well.

I would rather create my own phrase rather than putting any more mileage on "googling" than it already has. A phrase I remember having used before to avoid referring to any search engine in particular is "cyber safari."

E.g. I had always wondered when and where the phrase "larger-than-life" was born, so I embarked upon a cyber safari.

I'm glad you're asking, because we really need to stop using "google" as a generic verb. It's a toxic development that reinforces Google's unhealthy search monopoly in people's minds. But enough about that. I favor "web search," or just "search" when context makes clear that I mean an internet search.

Sadly, there's no term I know of that's as smooth and easy as "google." A phrase like "web search" still sounds to me like a noun more than a verb, requiring the use of chunkier expressions like "conducted a web search" or "searched the web for—". Alas, I have no solution to that. But the one thing I won't do is use google as a verb, even if Google is the search engine I happened to use in a particular case (which is unlikely).

Sometimes I say this:

You can use your preferred search engine to find more information on XYZ.

Or this:

You can find this information here as well as through your favorite search engine.

I decided on the phrasing "your preferred search engine" six years ago because I thought it would be clear enough that I mean "google, but not necessarily with Google" (and maybe it's also clear that I intentionally avoided saying "google"? not sure).

I don't remember exactly what other "your preferred X" constructions I had already seen at the time, but today, I typed "your preferred" into Google, and there are many suggestions that come up. The top suggestion is "your preferred pronouns" which I think only became a common phrase relatively recently, but there are other suggestions that must be older, such as:

• "your preferred phone number"
• "your preferred time"
• "your preferred time slot"

And when I typed "use your preferred" into Google today, there were only three suggestions, but still:

• "use your preferred name"
• "use your preferred payment method"
• "use of your preferred learning style"

So at least "your preferred X" is not a new kind of phrase.

Additionally, you can take out "preferred" or "favorite" and just say use a search engine.

Even when I pay close attention to the tutorial, I sometimes end up using a search engine just to figure out basic actions.

Unfortunately, these are all fairly clunky, I haven't seen anyone else say anything like these yet, and I don't know how this sounds to other people...