While not essential to reading the books on this list (we only suggest English translations here), learning ancient Greek and Latin has many pragmatic benefits and opens up a world of classical literature.
A highly-recommended series of books on self-learning ancient Greek are the three books of the "Reading Greek" series first published by the Joint Association of Classical Teachers in 1978. They are designed to be read in order: starting with 1 Text and Vocabulary, followed by 2 Grammar and Exercises, and finally 3 An Independent Study Guide.
Many of the negative reviews online find the Reading Greek series to be poorly organized in the older editions and recommend Mastronarde's 4 Introduction to Attic Greek as an alternative. However, the newer editions of Reading Greek seem to be better organized. The biggest difference is that the emphasis of Reading Greek seems to be more on reading than grammar, while Mastronarde's book emphasizes the grammar more.
For a beginner learning Latin, nothing beats Hans Ørberg's Lingua Latina Per Se Illvstrata series. It starts you off reading Latin from simple sentences and builds up to fairly complex ones; enough to follow along with many original Latin texts. It consists of two parts to be read in order: 5 Familia Romana and 6 Roma Aeterna.