The Help Center really only covers the most basic stuff (the list of privileges is pretty good though). You can get better results by searching Meta. It's worth noting that you'll get more (and often better) results on the Main Meta or Meta Stack Overflow.
They're working on improving search as part of the Channels project on Stack Overflow:
I can vouch for the fact that search work will be a big part of this project. We are aware of shortcomings with our search and will be improving it as part of Channels. — Haney♦
I'm not sure when any of that is going to happen, so I've listed some stuff you can do already to get better results.
There are a few places you can search built right in. Some people swear that the Ask a question page is a great way to find things, but you can't use any of the search operators. While it does support search operators, I don't like using the "close as duplicate" search because it only searches for "answered" questions. I like using the regular search bar in the corner (which is what I'm giving advice about here).
First of all, make sure you're on the best tab for what you're trying to do. If I think I'm looking for a popular post, I might use the voted tab, for example. The other thing to use is the right search operators. Check the help center for a full list of search operators. Here are some worth noting:
- Quotes help you find a specific word or phrase. Quoting a word will keep other "conjugations" from matching (e.g.
"word" will not match words).
- Exclude a word, phrase, or tag with
-"not this phrase",
- Search questions or answers with
is:a respectively. Searching for answers is helpful when you think a single-word-request is a duplicate, for example.
- Search the title or the body with
body: respectively. The title operator is pretty useful in particular because important stuff ends up in the title.
- Tag search with
[brackets]. I have not have much luck with this on ELU (because too much is tagged only grammar). On some sites, it's really useful (for example, programming language tags on Stack Overflow).
- Find questions that are closed or not with
Some people prefer to use a search engine like Google. One advantage is that it will return results that use similar words (e.g. you used the word beginning, you get matches for start) and misspellings. Or, you can use quotes around a word or phrase to get an exact match.
Another good tip is to use the
site: operator to restrict the search to one site. For example, like this:
The links at the bottom of this page will have more information on search operators or advanced search.
Searching all of SE
It's also worth noting that https://stackexchange.com has a custom Google search bar that lets you search the entire network, including metas. This is the easiest way to search only SE sites.
Ultimately, my advice is to not limit yourself to one method of searching. In fact, it's often appropriate to mix search operators while searching (you can also use the same search operator multiple times in a single search). Experiment and find what works best for you and what you're trying to find.
Also, a good proactive measure is to edit, edit, edit! Make sure the tags are accurate, make sure everything is spelled right, and make more drastic edits when it makes the post easier to find.