All the area libraries are part of the NY State Clinton/Essex/Franklin library system. CEFLS has all the libraries in the region on the same catalog system. This enables much more inter-library lending, reducing the funds needed for multiple copies of the same book for each library. You can even search for and order books online. It works very well if you know what you want, but going to the library to browse the shelves of say, new acquisitions, can still be enjoyable too.
All the CEF libraries offer some sort of internet access during open hours - check with the librarian for details. They can be over crowed during peak times and you will find the network service to be congested and slow. It you need net access when the libraries are closed try the KeeneLink wifi services at 5 locations in Town - they are very cheap and very fast.
The Keene Valley Library is open Tuesday, Thursday and Saturday, except for lunch hour. In July and August it is open more often and offers a Monday night lecture series that is a long-running summer favorite in the valley. Check it's calendar for details. The archives of this library are extensive, well organized and regularly staffed with experts able to help you, but call ahead (518-576-4335) to explain your area of interest and make an appointment. This library is operated almost entirely with donations from patrons and income from its endowment fund. It is a very nice space to enjoy quietly.
The Keene Public Library is normally is open all day, including lunch hour, on Mondays and Wednesdays and also Saturday morning. Check the web site before you go to be sure it is open. The Keene Food Pantry is located in the back of the building (open Wednesday mornings). This library's budget it is a line item in the Town budget, paid for by local taxes. It is a funky space, a bit rumpled. Some people love it and someone recently told me they think it is classic example of wasted tax money. "Why do we need two libraries?" is a long running question.
The Wells Memorial Library, in Upper Jay, also has a loyal group of users from Keene. They say it is more focused on books. Lastly, the Lake Placid Library on Main Street in LP has Keene users who rave about the place. It has sitting areas with very nice views of Mirror Lake and it is open every day except Sunday.
Wait...