For a basic search engine to research newspaper articles, I've found Google's news archives to be helpful. It's a free search, which will direct you to either a free article, or to the first paragraph of an article in those cases where the paper has pay-to-view archives (in some cases, the search will direct you to a newspaperarchive.com page; in those cases, if you have a subscription, simply log in and away you go).
In most cases, even if the article is pay-to-view, the fee is only $1.95 to $2.95. The advantage to Google's search is that it will also direct you to online news columns that were never available in print. Another helpful tool is the "timeline" feature, which shows the articles in a graph form so its much easier to find the proper time period you are researching--it also shows you when there where several articles written on the same subject, which has helped me to quickly discover if a case I'm researching in one time period may have also been discussed at length in another time period (most likely indicating a renewed interest in the case due to new clues or the case being resolved).
As for pay-site searches, I've long been a fan of newspaperarchive.com. I know some people don't like it, and its not for everyone---depending on the individual interest. Many papers have discovered there is money to be made from archive searches, so they don't allow access to their archives for recent years. So, in many cases, there won't be articles on NA from the past couple of years for many papers. And, there are time periods and areas from which archives simply aren't available at the present time. The thing to keep in mind is that new papers are being added daily, so I've found NA to be worth the subscription price (especially since my area of interest is the 70s and earlier).
Another thing to keep in mind with NA is that the papers are often hard to read, so the search engine may not return on the first search parameters you enter. I've found it helpful to alter the spelling on certain words, and sometimes I'll get a hit that I didnt get before. Its trial and error in some cases.
Also, different search terms will return different articles. Searching "girl missing" will return different articles than "missing girl" and "girl reported missing". Though all may return some
of the same articles, none will return the same
Don't be overly specific with the area that the incident you are researching occurred. You may be looking for something that happened in 1945 in Kansas; don't limit your search to Kansas. Keep in mind that most stories went out "over the wire" via AP, UP, and INS, so the info you're looking for might actually be found in an Iowa paper, or a Montana paper...
The most important thing is to be patient. Researching older cases takes time. Take good notes on what you find, and don't get discouraged if you don't find what you're looking for right away.