[...] in the end I decided I'd better not try any more systematic instruction. Today I'm sorry I didn't, because I can see how one-sided my alleged education is. I am abysmally weak in modern languages, and mathematics, and in certain phases of history (such as that of the European Continent in mediaeval and modern times)—while my absence of training in economics and sociology is really a deplorable handicap to me in my efforts to understand the trend of these tense times, when so much of the motivation of nations, as well as their internal problems, depend almost wholly on complex economic consideration. If I seem to some people to overvalue learning, it is because I am so damned ignorant myself. We appreciate what we lack ourselves. What has redeemed me from a still worse ignorance is a fairly lively curiosity about things in general; which has led me to read up on various subjects, and to argue at great length with people who know more about things than I do.

•H. P. Lovecraft to Robert E. Howard, 29 Mar 1933, Selected Letters 4.177, A Means to Freedom 2.580

First, a bit of background: in A Dreamer and a Visionary 43, S. T. Joshi writes:

He attended the Slater Avenue school in 1898-99, but was then withdrawn; he resumed schooling there for the 1902-03 school year, and was withdrawn again. he adds that 'In 1903-04 I had private tutors'. We know of one such tutor, A. P. May, although Lovecraft did not have a very high opinion of him.

In the 1904-05 school year, Lovecraft attended Hope Street English and Classical High School. (Joshi 49-50)

It is worth studying what courses Lovecraft actually took during his three years at Hope Street. His transcript survives, and it is full of interesting and suggestive information. The school year lasted for thirty-nine weeks, and most of the courses Lovecraft took covered an entire year; occasionally he took courses lasting only one term, either nineteen or twenty weeks. (In the following enumeration, classes are for thirty-nine weeks save where listed.) Numerical grades were issued; an 80 represented a Certificate grade, 70 a passing grade. During the 1904-05 year, Lovecraft received the following grades: Elementary Algebra (74), Botany (85), English (77), Ancient History (82), and Latin (87). There is not much that unusual here, except the surprisingly low grade Lovecraft received in English.

Lovecraft returned to Hope High in September 1905, but his transcript states that he left on 7 November of that year, not returning until 10 septemeber 1906 (presumably the beginning of the 1906-07 school year). This is no doubt the period of his 'near-breakdown' of 1906. There is not much evidence as to the nature of this illness. [...] When Lovecraft returned for the 1906-07 school year he received high marks in English (90), Plane Geometry (92), and Physics (05), and good marks in several other subjects, including Drawing, Latin, and Greek. The ominous thing here is the continuing low marks for Algebra (75).

In his final year at Hope High (1907-08) Lovecraft took only the following: Intermediate Algebra (ten weeks) (85); Chemistry (95); Physics (95). Here the interesting this is his retaking of Algebra. Lovecraft himself remarks: 'The first year I barely passed in algebra, but was so little satisfied with what I had accomplished that I voluntarily repeated the last half of the term.' There is a slight inaccuracy here, since it was not the Elementary Algebra of his first year that he retook but the Intermediate Algebra of the second year; and he does seem to have finally achieved a better grade this time.

The transcript states that Lovecraft left on 10 June 1908, presumably at the end of the term, since he is recorded as having the full thirty-nine weeks of chemistry and physics. But he clearly did not receive a diploma, and indeed it is evident that he has only finished the eleventh grade—or perhaps not even that, since he anomalously took only two full courses during this third year. He would surely have required at least another full year of schooling to qualify for graduation.

Most of Lovecraft's education while very young was done by "private tutors" - we don't know much about what that would have entailed for Lovecraft at age 13 or 14 (1903-1904), common subjects would have included English, Greek, Latin, History, and Mathematics, but we don't really know what the tutor's specialties were - and probably consisted largely of lessons from a text book and various exercises. Given his later regrets and lacks, we can probably assume that modern languages (French, German, Italian, Spanish, etc.), economics, politics, etc. didn't form a part of this curriculum. Given his family's declining finances, these probably weren't the best available either. We also know he apparently took a correspondence course in chemistry around 1909. (Joshi 64-65)

Lovecraft's real education at home, as in later life, came from his voluminous reading, often in the family library (which included medical textbooks from an uncle, and dated 18th-century volumes) and the Providence Local Library. One example of such "self-study" in his letters concerns his sexual education:

When I was six or seven I was of course curious about the allusions which I did not understand in adult books, and about the prohibitions imposed by elders upon my conversation. Being of a scientifick and investigative cast, I naturally followed up the mysteries step by step in encyclopedias and other books—for with my temper no one dared seriously restrict my reading. Ending with the medical books of my physician-uncle, I knew everything there is to be known about the anatomy and physiology of reproduction in both sexes before I was eight years old; after which curiosity was of course impossible. The entire subject had become merely a tedious detail of animal biology [. . .] (Selected Letters 1.304)

Other than that, we don't know exact details. H. P. Lovecraft did briefly attend Sunday school at a young age (exact accounts vary, starting either at 5 or 7 and ending at about 12):

How well I recall my tilts with Sunday-School teachers during my last period of compulsory attendance.! I was 12 years old, and the despair of the institution. None of the answers of my pious preceptors would satisfy me, and my demands that they cease taking things for granted quite upset them. Close reasoning was something new in their little world of Semitic mythology. At last I saw that they were hopelessly bound to unfounded dogmata and traditions, and thenceforth ceased to treat them seriously. Sunday-School became to me simply a place wherein to have a little harmless fun spoofing the pious mossbacks. My mother observed this, and no longer sought to enforce my attendance. (Selected Letters 1.110-11)

Lovecraft's recollections might be colored a bit, but religious instruction generally didn't "stick" to him.