The ease or difficulty in finding statistics varies a lot, depending on what statistics you are looking for. Be sure to carefully evaluate the authority of the organization offering the statistics, and if there is any information about how the statistics were collected.
"From the Horses Mouth" - Look for Statistics from a Particular Government Agency or Organization
But I Don't Know Which Horse...
The library has some major compilations of statistics, such as Statistical Abstracts. See the Major Compilations tab. Try keyword searching those. If you find what you are looking for, or at least are fairly close, look at the source information at the bottom of the table. This will tell you the name of the government agency or organization that cares about that data. Google that name and delve into the website.
Keep an Eye Out as You Are Researching...
As you are finding background information or articles on your topic, take note of any relevant statistics. Some scholarly journal articles may have done detailed studies on the issue and generated statistics. Or a news article may mention a juicy statistic to jazz up the article.
Note the source -- you may have to dig into the references if it's a scholarly source, or look for who was interviewed in the news article.
Let's say my issue is animal shelters.
Animal shelters are at the city or county level, and there are a mix of governmental ones and nonprofit shelters. (If you didn't already know this from prior experience of adopting a rescue or figuring out who might have your lost pet, hopefully you would learn this quickly via the background information you looked at at the beginning of your research.)
Locally, Spokane County has SCRAPS, Spokanimal, and the Spokane Humane Society. I will want to check of of their websites to see if they have statistics on how many animals they house, adopt, etc. each year.
NOTE: If you don't find what you're looking for online, don't just give up. The next step is to find contact information and call them and ask if they have the statistics you want. Small agencies/organizations may not have much in the way of resources to put lots of information on their website, but the data is collected and kept in a filing cabinet.
If we wanted to know a broader question on how many American own a pet, while it may seem like a common enough question that it would likely be in a major statistical compilation, like the Statistical Abstract. But no, they only have how much we spend on pets.
So try google, but be very picky about where they got the data. By looking through the references of likely pages, it turns out the American Veterinary Medical Association collects information on household pet ownership. This website summarized and linked to the American Veterinary Medical Association's 2018 200+ page report. Says 56.8% of households owned a pet in 2016.