You may also find quantitative results concerning error term from prime number theorem in the article S. B. Stechkin, A. Yu. Popov, “The asymptotic distribution of prime numbers on the average”, Uspekhi Mat. Nauk, 51:6(312) (1996), 21–88, english translation is published in Russ. Math. Surv. 51 (1996), pp. 1025-1092.
