Keep your searches simple - If you're looking for a specific thing or concept, just enter as much of the name as you remember. Most searches aren't going to require search operators or special syntax. 

Use words you think are most likely to appear on the page - Search engines match the words you enter in the search box to words on the pages it searches. For example, you wouldn't search my head hurts, you would search headache. 

Use as few terms as possible - Since almost every word in a search is used, the more words you enter, the more limited your search will become. So, using too many words may limit your results. Alternatively, Google will sometimes choose to ignore certain terms that will give you results that aren't relevant. So for example, weather Lebanon Oregon  will give you better results that what is the weather report for Lebanon Oregon. 

Choose descriptive words - The more specific words you use, the more likely you are to get relevant resutls. Words like "article," "website," or "info" aren't very descriptive and won't be very helpful. Also, keep in mind that even if a word has the right meaning, but you're using it in a way that's not particularly common, your search won't match the pages you need (remember, we want to use the words that are most likely to appear on the page!). For example, a search for animal sounds and a search for animal ringtones will give you two very different results. 

Quotes: Quotes allow you to search for an exact word or set of words. But make sure you only use this if you're looking for a very specific word or phrase, otherwise you'll exclude a lot of results. 

Ex: Searching Pumerantz Library will retrieve results that include the word Pumerantz, the word library, and the phrase Pumerantz Library. Searching "Pumerantz Library" will only retrieve results with that exact phrase. 

Dash: If you add a dash before a word, it will exclude all results that include that word. This is especially useful for synonyms. Also, if you add a dash before a website, you can exclude results from that specific website. 

Ex: Searching jaguar speed -car will show you articles about jaguars that don't include the word car. Similarly, searching pandas -site:wikipedia.org will show you articles about pandas that aren't from Wikipedia. 

Site: Adding site: to your query allows you to search within a specific website or domain. 

Ex: The search string library site:westernu.edu will show you all the pages on westernu.edu that contain the word library. 

Link: If you add link: to your searches, you can find pages that link to other specific pages. 

Ex: link:westernu.edu will show you all the pages that link to westernu.edu

Related: Adding related: to your search allows you to search for sites that are similar to a URL you already know. 

Ex: If you search related:westernu.edu, you get a list of results that are similar to that website - in this case, a list of other health sciences schools. 

*: Include * as a stand in for words in a search that you don't know. You can even use it with quotes to find variations. 

Ex: a * in the hand is worth * in the bush. 

Or: If you want to search for pages that have just one of a few words, just add or to the search. 

Ex: westernu graduation 2018 OR westernu graduation 2015 only includes results that have the words, westernu, graduation, 2018 and 2015. All other results are excluded. 

..: Separate numbers with two periods without spaces to see results that cover a specific number rage - such as dates, prices, and measurements. 

Ex: computer price 400..800 will give you Google shopping results for computers in that price range. Most of your actual search results will be articles/blog entries about computer prices. 

For more information, you can check this list of search operators and other techniques for refining searches from Google Support.