Standard Search. You'll see the standard Twitter Search box in the upper right-hand corner of any page at Twitter, and to make better use of the standard search, here are some search operators you can use (source)... I'm not really sure how those "attitude" operators work, but you can give them a try! And if there is a search you are going to do frequently, you can save the bookmark in your bookmarks folder.
More Options. When you are looking at your search results, you can access a wide variety of additional options, as shown in the screenshot below:
For example, in addition to saving searches by bookmarking them, Twitter allows you to save searches which will show up as selections in the Search box dropdown.
Are there some searches you think you might do repeatedly at Twitter? Do that search, including whatever search parameters help you get good results, and then save the search!
So, use the search and advanced search features to look up some people and topics of interest to you. You might find some new people to follow as a result!