An excellent review of Open String Field Theory may be found in the paper "Analytic methods in open string field theoryAnalytic 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.
The following detailed lecture notes by Washington Taylor and Barton Zwiebach are also of pedagogical in nature: "D-Branes, Tachyons, and String Field Theory": [arXiv:hep-th/0311017]. Carlo Maccaferri has a succinct review of the subject which gives a broad picture and clearly introduces the CFT formalism : Basics of Open String Field Theory
Another useful review may be the one by Ehud Fuchs and Michael Kroyter: "Analytical Solutions of Open String Field Theory" hep[arXiv:hep-th/0807.47224722]. 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": hepBatalin-th/0301094
The BVVilkovisky (BV) formalism - which is useful for closed string field theoryconstructing Closed String Field Theories - is discussed in the articlecontext of Open String Field Theory by Charles B. Thorn if you are interested in that (although I haven't managed to go past the early few sections)this review article: "String Field Theory"
Carlo Maccaferri has a succinct review of the subject which gives a broad picture and clearly introduces the CFT formalism : [Phys.Rept. 175 Basics of String Field Theory http://www.ulb.ac.be/sciences/ptm/pmif/Rencontres/ModaveII/Carlo.pdf
And if you are interested in the(1989) 1-101]. The Berkovits formalism for superstring field theory, the following review might be useful is discussed in : "Review of Open Superstring Field Theory" by Nathan Berkovits - hep[arXiv:hep-th/0105230.0105230]