An excellent review of String Field Theory may be found in the paper "Analytic methods in open string field theory" by Yuji Okawa: https://inspirehep.net/record/1226324?ln=en. It develops the CFT formalism and presents the Schnabl solution and the associated solutions with detailed calculations.
Another useful review may be the one by Ehud Fuchs and Michael Kroyter: "Analytical Solutions of Open String Field Theory" hep-th/0807.4722. It discusses the CFT and the oscillator formalisms for covariant string field theory along with the Sen conjectures.
The following lecture notes by Washington Taylor are also of pedagogical in nature: "Lectures on D-branes, tachyon condensation, and string field theory": hep-th/0301094
The BV formalism for closed string field theory is discussed in the article by Charles B. Thorn if you are interested in that (although I haven't managed to go past the early few sections): "String Field Theory"
Carlo Maccaferri has a succinct review of the subject which gives a broad picture and clearly introduces the CFT formalism : Basics of String Field Theory http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveII/Carlo.pdf
And if you are interested in the Berkovits formalism for superstring field theory, the following review might be useful : "Review of Open Superstring Field Theory" by Nathan Berkovits - hep-th/0105230.