I have a feeling that the vast majority of people on Twitter are not aware of how to take advantage of Twitter's search capabilities. I've had people assume I spent a long time finding something when it took seconds

I'll explain how to use some of options with Twitters search.

Just type it into a search box.

Ever want to find a tweet from someone in particular?
from:username --- for all tweets from that person.
After the username you can enter search terms to get tweets from that person including those words.

from:realdonaldtrump cruz

Trying to find a tweet that was sent to a person?
to:username --- for all tweets mentioning that person.
Like with from, you can add search terms. You can also combine these searches to find all tweets from someone and to a particular person.

from:texantruth42 to:betoorourke

Some filters for searches:
With many of these filters you can add a minus sign before filter to get the opposite results. (-filter:replies to get no replies)

filter:verified will give you search results only from verified users.

filter:replies This will give you only reply tweets.

filter:retweets This removes retweets from search results.

filter:links To see only tweets with links

filter:images For only tweets with images

filter:media For tweets with video or images

filter:native_video tweets with video uploaded to Twitter

since:2018-01-01 until:2018-05-15 Tweets from within a time span of 1-1-2018 to 5-15-2018

min_faves:10 will show you tweets with 10+ likes. You can use whatever number you wish.

near:houston,tx within:10mi See tweets from a certain area

These can be used and combined to allow you to find information/tweets very quickly. You don't have to scroll down someones profile trying to find something. This works for the app and the website.

On the website there is this that you can use https://twitter.com/search-advanced 

Some additional search tips:

Operators

"United States" would find the exact phrase.
Dr -Pepper would find Dr without Pepper.
hot OR cold would find tweets with hot or cold.
lang:de for tweets in German
lang:nl for Dutch
lang:es for Spanish.. etc..

Want to find news?

add
filter:news
to your search term(s).
It will find news with the tweet containing that search term.

Want to find tweets with a minimum amount of retweets?

min_retweets:10 search-terms

I didn't include this one previously because I found it wasn't very reliable when mixed with the from:username filter. It seems to work well outside of that.

This is for searching for tweets from people on a list.

dogs list:MSNBC/correspondent-list

This one you would just want to bookmark after you search it or add it to a saved search. You have to manually go to the profile to find the list name to set it up. Name is in URL.

If you want an easily printable PDF of the search tips from my thread here it is:

http://docdro.id/bQtM8ah 

There is a download icon in the top right of the page to save it to your device.
It saves as Search.pdf

One last filter that was brought to my attention.

filter:follows Shows you only the results from people you follow. https://twitter.com/BramStolk/status/1001471459389202432