Many plays and monologues are available online, either through library subscriptions, or for free. Because plays from before 1924 are in now in the public domain, older works can often be found freely online. Some more recent works are made available online by the authors to promote performances of their works, which would require paid licensing. If you Google a play and find the full text online, make sure you're looking at a legal and reliable copy!
Through the library:
Through the web: