When abbreviating people’s names into a string of letters, separate the letters with a period and no space or with a period and hair space. Thus prefer either J.R.R. Tolkien (no space) or J. R. R. Tolkien (hair space) to J. R. R. Tolkien (full space). For ease and simplicity, this dictionary uses the unspaced style.
For all other abbreviations, English maintains no consistent rule on punctuation. Usually, when the abbreviation is an initialism (an abbreviation read letter-by-letter), the abbreviation is punctuated (e.g., U.S.A., C.I.A.); when it’s an acronym (an abbreviation read as a whole word), especially if it’s rendered in lowercase, it is not punctuated (e.g., UNESCO, fubar, laser). But exceptions are common if not standard: CIA is far more popular than C.I.A., and mph better than m.p.h. Whatever you choose, keep each abbreviation consistent within your writing. Don’t write r.p.m. in one paragraph and rpm in another.
Define lesser-known abbreviations before using them. Most everyone knows what the U.S.A. is and does not need it defined, but any less-universal abbreviation should first be written fully before it is abbreviated: If you need your medical release of information (R.O.I.), ask for the R.O.I. documents during your next doctor’s visit.